Cosmology

[YesNo]

You should be able to see to the bottom of the well.

I have been using Google Sheets to gather data about primitive roots as well as checking some papers online to get a feel for the problem. Since I need to show that 2 is a primitive root for infinitely many primes, I will have to find some way to use the information about a finite number of primes (assume all of them) for which 2 is a primitive root and reason from there that there must be another one.

[desiresjab]

You should be able to see to the bottom of the well.

I have been using Google Sheets to gather data about primitive roots as well as checking some papers online to get a feel for the problem. Since I need to show that 2 is a primitive root for infinitely many primes, I will have to find some way to use the information about a finite number of primes (assume all of them) for which 2 is a primitive root and reason from there that there must be another one.

I see what you want--something that proceeds along similar lines to Euclid's proof of the infinitude of primes.

Is this a proven proposition--that 2 is a primitive root for infinitely many primes--or someone's unproven conjecture?

Was this something to do with Artin's conjecture? How quickly I forget except what is in my tunnel vision.

[desiresjab]

Okay, now I have refreshed. That type of simple appearing yet intractable problem is typical of number theory. One sometimes is astonished that certain propositions which are so simple go unproven for so long. No doubt, many a doctoral dissertation has beat its head against your particular problem, which is indeed Artin's conjecture.

[Dreamwoven]

Good luck to both of you in your search.

[YesNo]

Thanks, Dreamwoven!

Yes, it is Artin's conjecture, or part of it. A lot of number theory books are available as pdfs from the internet. There are more than I have time to read. Luckily for me I don't have to read all of them since they repeat themselves, just understand a few.

[desiresjab]

Eisenstein has done something extraordinary. His proof actually has nothing to do with QR, other than a method for building the right exponent to go on -1, though it does involve the primes p and q. It looks at the ratios of p and q when the prime in the numerator is multiplied by successive even numbers under the modulus of the other, with a chop function appended. In other words a function that always rounds down instead of moving to the nearest value. It would have been sufficient to find the correct parity under any circumstance, but Eisenstein is more exact than that, producing the exact exponent on -1 as the number of lattice points in the prescibed regions of his p by q rectangle. Only the (p-1) by (q-1) part interests him, containing the interior lattice points of the larger rectangle, and then only those with even coordinates.

This proof is wonderfully clever. The downside is that it will not reveal any deeper properties of numbers that help elucidate why QR works. Deeper investigations might uncover why it works. It proves what it proves--that Eisenstein's method will always find the right exponent for -1 in the Legendre symbol.

[YesNo]

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

[desiresjab]

Is the computation done in polynomial time in terms of the number of digits of p and q or something slower?

Uhhhh....I am not sure I understand the question.

Finding a quadratic residue of a given number no matter how large, should be a P time exercise. You may be referring to something like RSA encryption or Diffe-Hellman key exchange. Various encryption systems are based on particular aspects of number theory laws. It can be quadratic residues, primitive roots or mod inverses they use to "conceal" the message.

In reverse the problem gets nasty real fast, as in NP nasty. It is easy to give some quadratic residues of a number q. But given a quadratic residue, it is impossible to find q in ploynomial time, if the numbers involved are long enough.

Currently, it takes numbers of about four thousand digits length to get your bank account information encrypted securely. If some country or individual had the power to quantum compute, it has already been mathematically proven they could break our present codes in seconds.

[YesNo]

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

[desiresjab]

That answered the question. I was wondering if there were a computation problem still unsolved. I suppose a quantum computer runs faster because of a potential parallel processing involved. Wouldn't a network of computers working in parallel be able to simulate such a computer?

That is the same question I have had. I have a hunch the answer would be "yes," provided that we link enough silicon computers together to fill the solar system, or some other great volume of space.

I went to town today and forgot graphing paper. If I graph out some p's and q's Eisenstein style, another key to QR may pop out visibly or algebraically.

I have some other studies I am stalling right now because I cannot let go of QR when I am so close. Like one of those movie bounty hunters who has pursued a particular fugitive for a long time, I cannot go for coffee now that the fugitive has been sighted. However, at my age I find I need two days rest after pursuing the most intense thinking for one day. Total forced mental focus was something I learned in math and then adapted for dummies from reading about Newton and Tesla. Now, in math I have to force it, because it hurts me to concentrate that hard on x, y and z in a foreign medium, whereas in a field like fiction writing I can stay immersed indefintely without forcing myself to, indeed without ever becoming conscious of the need to force myself to do anything.

The creative process is so much more enjoyable while it is happening, but when titantic struggles with x, y and z are over and have resulted in a surrender without terms, I find my picture of the universe is altered, and my picture of myself, as well. In this medium of math I am no natural, but I am a zealous convert.

[Dreamwoven]

I am lost in this discussion, never was any good with maths. This discussion of time-warps and the past and future of Space-Time may be of interest: http://www.space.com/31495-space-time-warps-and-black-holes.html?cmpid=NL_SP_weekly_2016-1-04

[desiresjab]

I finally reeled Eisenstein in all the way. Of course that does not mean I understand Eisenstein as well as Eisenstein did. I feel like champagne anyway.

[YesNo]

Interesting survey article on spacetime, Dreamwoven. In particular this quote:


"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikenaga/number-theory/quadratic-residues/quadratic-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

[desiresjab]

Interesting survey article on spacetime, Dreamwoven. In particular this quote:


"It might be that space-time at very short distances takes yet another form and perhaps is not continuous," Amendola said.

Congratulations, desiresjab! I was reading this about quadratic reciprocity. It is a very elementary summary of it: http://sites.millersville.edu/bikenaga/number-theory/quadratic-residues/quadratic-residues.html

The part I was interested in was if a prime is a primitive root for another prime it would have to be a quadratic nonresidue. I haven't tried to understand Eisenstein's proof.

You are welcome. As usual, one wonders why he didn't see it sooner. I guess it is the leap from the quadratic to the linear. -1 is a dummy base, only good for determining if the exponentiated value is positive or negative. His whole lattice graph is linear, yet it explains a quadratic law. People like Eisenstein are not really human, they just share some genes with the rest of us.

I am going to toy around with making a new encryption system. If nothing else, I will learn the present systems better. I have to come up with an appropriate function.

* * * * *

There is no guarantee that quantum theory and Einsteinian physics are uniteable. Parts of Einstein are already dated, Ed Mitchell has said. What if never the twain shall meet?

[YesNo]

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

[desiresjab]

If those quantum computers ever happen beyond a few qubits we may need a new encryption system.

Yes, and it is hard to even imagine what it might be. I have been kicking around some ideas for a system that would be easily patentable. The normal trick is to multiply two huge primes p and q together to produce n, which will be part of the modulus. Pick a number e relatively prime to n as an exponent to encrypt the message, as in Me. Then you find the inverse of e (mod φ(n)). It is this e-1 which will be used to untangle the message on the other end. You cannot find φ(n) without knowing the factors of n. That is the immense diffculty they impose on hackers--they have to find φ(n) to get anywhere.

I am looking for a system that does not rely on factoring. Encryption is one area of math with a big, big future. I have a few wild ideas.

[YesNo]

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

[desiresjab]

Maybe using larger primes will keep the current methods going until something better comes along.

I'm still putting the pieces together on the Artin conjecture. A simpler question would be "Given a number m, are there infinitely many primes p for which m is a quadratic nonresidue?" This would be a larger set since a quadratic nonresidue does not have to be a primitive root such as 8 or 12 mod 19 unless I calculated it wrong.

One aleph is as big as the next aleph.

[YesNo]

I thought Aleph one was strictly bigger than Aleph null.

[desiresjab]

I thought Aleph one was strictly bigger than Aleph null.

Otherwise known as Aleph nought, as well, the "smallest" infinity. They are the same entity, all of them, as far as I know.

To make all possible sets from a set of size n, take 2n. The next set after aleph, is 2 raised to the aleph power. You can repeat the process over and over getting bigger sets. No one knows which of these powers has the power of the continuum.

There is an infinite heirarchy of infinities, each theoretically greater than the next, but we only have examples of two kinds. One can be thought of as the counting numbers, and the other is the uncountable points on a line, i.e. the irrataional numbers, more specifically the transcendental numbers.

The former do indeed have the cardinality of aleph, and the transcendental numbers may have the cardinality of the continuum. I believe the latter is not known for sure. Kronecker would certainly disapprove.

[Dreamwoven]

Impressive knowledge!

[YesNo]

The set of functions over the reals should make a larger infinity than the reals. Which is larger than the integers or rationals. But Kronecker may be right that all of that probably doesn't matter.

[Dreamwoven]

You guys have learned a lot of stuff that I never did in my London school in 1956. GCE Ordinary Level. Never did math beyond that.

[YesNo]

I wish I knew some physics and chemistry. What I know of quantum physics can be traced to someone posting something about many worlds on Lit Net and then going off to the library or the internet to try to make sense out of it. I think I know enough about many worlds at the moment to be able to reject it. The same with black holes, but I am less sure about black holes than I am about many worlds. I didn't even know about the big bang (except as some vague idea) until someone posted that the universe started from "nothing" including space and time. I got to the library as soon as I heard that. One of the most shocking moments of enlightenment was when I heard about a youtube video claiming that we never put a man on the moon. It took two days to get over that and now I'm convinced.

So now I figure if they can put man on the moon, I can solve the Artin conjecture or desiresjab can come up with a new cryptography method.

[desiresjab]

The set of functions over the reals should make a larger infinity than the reals. Which is larger than the integers or rationals. But Kronecker may be right that all of that probably doesn't matter.

The set of functions over the reals will not be of greater cardinality than the reals themselves. Even the transcendentals are part of the reals, and are of course as large as the whole set, in the same way that the set of even numbers is as large as all of the rationals. Cantor proved this. All that matters is whether you can map elements from one set to the other with a one-to-one correspondence. You could produce this correspondence between sets seeminlgly so sparse as the square numbers and one as dense as the rationals. They are both aleph nought in cardinality.

[desiresjab]

I wish I knew some physics and chemistry. What I know of quantum physics can be traced to someone posting something about many worlds on Lit Net and then going off to the library or the internet to try to make sense out of it. I think I know enough about many worlds at the moment to be able to reject it. The same with black holes, but I am less sure about black holes than I am about many worlds. I didn't even know about the big bang (except as some vague idea) until someone posted that the universe started from "nothing" including space and time. I got to the library as soon as I heard that. One of the most shocking moments of enlightenment was when I heard about a youtube video claiming that we never put a man on the moon. It took two days to get over that and now I'm convinced.

So now I figure if they can put man on the moon, I can solve the Artin conjecture or desiresjab can come up with a new cryptography method.

Good sir, none of us knows enough. The day belongs to those who seize it. Here I am at sunset trying to seize the day. Far be it from me to discourage any research. I wish you luck. Artin's conjecture is really formidable. It is one of those questions that has attracted the best quality of research. You will need to become something of an ace at modular arithmetic, since the conjecture deals with that branch, not normal algebra. The more tools you have the better you can understand what has already been done. You cannot scale Everest without some climbing gear. First, be very sure of what reciprocity means.

The guy in the link below helped me a lot with his articles. After his article on QR, the next article on biquadratic reciprocity is missing, but the ones after that are already written and posted. He ties Pytharorean triplets to reciprocity. This article gives a good idea of just how deep mere QR is, let alone general reciprocity. Hope I have not posted it before, but it is worth a read. I spent a long time on it and have probably read it ten or fifteen times. Anyone who has an easy time with this probably should have been a mathematician.

http://science.larouchepac.com/gauss/ceres/InterimII/Arithmetic/Reciprocity/Reciprocity.html

[YesNo]

Thanks for the link. If you have any more please post them. I realize there is a lot to get familiar with before I would even know that I solved anything at all.

The first step is to show for the number 3 that there are infinitely many primes for which 3 is a quadratic nonresidue. I am sure someone has done that already. Then I would need to know what the additional conditions are to guarantee that 3 was a primitive root as well.

[desiresjab]

Thanks for the link. If you have any more please post them. I realize there is a lot to get familiar with before I would even know that I solved anything at all.

The first step is to show for the number 3 that there are infinitely many primes for which 3 is a quadratic nonresidue. I am sure someone has done that already. Then I would need to know what the additional conditions are to guarantee that 3 was a primitive root as well.

Yes, that could be one approach. 3 is a quadratic nonresidue of all its 4n+3 residues, and a nonresidue of all its 4n+1 nonresidues. Primes have exactly as many residues as they have nonresidues under their modulus. That seems like a decent place to start nosing around for truffles. Can you smell that truffle?

[YesNo]

Let's see if I got this right. If I want to know if 3 is a nonresidue with respect to p, then I need to calculate (3|p)=31/2(p-1) mod p. If the value is -1 then it is a nonresidue. If it is 1 then it is a residue.

I might be able to get this information using quadratic reciprocity for p and q being odd primes. The formula is (p|q)(q|p) = (-1)1/2(p-1)*1/2(q-1).

Let q = 3, since that is the number I am interested in. Then 1/2(q-1) can be simplified to 1/2(3-1) = 1/2(2) = 1, so I can write the quadratic reciprocity rule as follows for q = 3.

(p|3)(3|p) = (-1)1/2(p-1)

Now what? What I am trying to find is (3|p), but with QR I also need to find (p|3).

[YesNo]

The set of functions over the reals will not be of greater cardinality than the reals themselves. Even the transcendentals are part of the reals, and are of course as large as the whole set, in the same way that the set of even numbers is as large as all of the rationals. Cantor proved this. All that matters is whether you can map elements from one set to the other with a one-to-one correspondence. You could produce this correspondence between sets seeminlgly so sparse as the square numbers and one as dense as the rationals. They are both aleph nought in cardinality.

I'll have to think about this one and maybe do some searching.

I suspect one could assume there is a 1-1 mapping between the reals and the functions over the reals and then construct a function that is not in that set using similar substitutions that Cantor did to show that the reals are larger than the integers, however, I wonder about the details.

[desiresjab]

Well, I think we do not know of a set larger than the reals, since the reals have the power of the continuum.

I will try to get to some specific answers to your other questions soon. I just got back online after service being out for a few days. You seem to be on the right track and understanding the subject. There is no reason I shouldn't be surprised at anyone who can do that.

2 and 3 have longer periods than, say, 7. Precisely what this means I am still figuring out.

The amazing thing about the Martinson articles is how they relate 4n+1 numbers to the hypoteneuse in right triangles. Then it goes on to speculate that 4n+1 primes are not really primes at all but an unnamed species of number. I had never heard or seen anything like that before.

[swathisharan]

cosmology is the best course as every one had craze about their beauty.

[YesNo]

The amazing thing about the Martinson articles is how they relate 4n+1 numbers to the hypoteneuse in right triangles. Then it goes on to speculate that 4n+1 primes are not really primes at all but an unnamed species of number. I had never heard or seen anything like that before.

I liked that about the article as well. I have been wondering why QR is so interesting. If it goes back to Pythagoras that would explain it. I assumed the 4n + 1 numbers were just a special subset of primes when I read that although the article suggests for some unknown reason that they are not prime numbers.

There was also a tension between Gauss and Euler that I was unaware of. Gauss seemed to prefer Fermat. And there was something deliberately hidden that associated Guass with Kepler.

Also I tried reading LaRouche's article and it didn't make sense. I have no problem with the economy being in the toilet, but I didn't understand why he thought it was.

[desiresjab]

I liked that about the article as well. I have been wondering why QR is so interesting. If it goes back to Pythagoras that would explain it. I assumed the 4n + 1 numbers were just a special subset of primes when I read that although the article suggests for some unknown reason that they are not prime numbers.

There was also a tension between Gauss and Euler that I was unaware of. Gauss seemed to prefer Fermat. And there was something deliberately hidden that associated Guass with Kepler.

Also I tried reading LaRouche's article and it didn't make sense. I have no problem with the economy being in the toilet, but I didn't understand why he thought it was.

Gauss was a teenager when he worked out QR. He did not know about the work of Euler, LaGrange and a few others in that area, according to him, and acheived his results independently. When he was almost done with Disquisitions, he found their work and set about cataloguing it along with all of number theory as it was known in Europe at that time. Then he launched his ship Disquisitions, one of the supreme texts of mankind, and one of the least heard of. One out of a million people reads it. Even fewer understand what they have read. Gauss is the guy to give you the law, but not the guy to help you understand it.

Euler was the guy to help you understand things, who would show you his failed attempts as well as his successes. Euler was a natural teacher. One knows full well that Gauss was accessible to only world class geniuses. Neither one of these guys stood before thronged classrooms of students, but the expository nature of Euler's writing style and the way an unusually fine personality was bolded forth is a beautiful thing to see in history.

Martinson's apparent odium for Euler is baffling to me. The way he discount's the entire latter half of Euler's career is shocking, but it certainly does create interest in the article. I would not accuse this gentleman of shock-jocking, though...ahem!

For a fact, Euler tried to untangle QR and failed only by a hair. Legrendre tried, too, and came close in a slightly flawed proof. Legendre was an ATG, but he lived in Gauss's shadow like Gehrig in Ruth's. The method of least squares was snatched away from him by history and Gauss.

The difference between Euler and Legendre is that Euler would make an ATG top top ten list in mathematics. On the Mt. Rushmore of mathematics, after Archimedes, Newton and Gauss, Euler is a powerful contender for the fourth spot.

Any tension between Euler and Gauss would have been based on the work alone, and strictly one-way, for Euler was dead by the time Gauss arrived on the world scene. Gauss was six when Euler died.

Like Newton, Gauss was a curmedgeonly neurotic, parsimonious with praise. He used a Latin word that praised Euler, but reserved for Newton the appellation of summa.

[YesNo]

Thanks for setting me straight on Gauss. I was beginning to think there was something wrong with the "turncoat" Euler as Martinson described him, but I realized I had no reason to trust Martinson's view either.

I downloaded Gauss' Disquisitiones in Latin. I should be able to use Google Translate to get around to the parts I might find interesting. I found a copy of Leonard Dickson's "Introduction to the Theory of Numbers". I figure I better know what is in a book like that.

[desiresjab]

The elementary number theory text I got the most out of was H.L. Davenport's Higher Arithmetic. I always find it useful to keep a number of texts around in case one fails me. I found the section on primitive roots to be very lucid.

I do not think QR goes all the way back to Pythagoras, or anywhere near. I am almost certain the theory was unknown to the Greeks. I do not know of any mention of it in the ancient world, even by Diophantus. Unlike the compass and straight edge problem Gauss solved from the ancients, QR seems to have been discovered by Euler in the west and proven by Gauss.

[desiresjab]

Wayward Fact #1:

-1 is never a quadratic residue of primes of the form 4n+3, and always a quadratic residue of primes of the form 4n+1.

Wayward fact #2:

The behavior of 2 with respect to QR is different from the odd primes, as we might expect, but still fits into the theory consistently.

[YesNo]

Yes, I suspect QR is a relatively recent idea.

Regarding the first wayward fact, -1 is p - 1 mod p. One can tell if -1 is a quadratic residue for odd prime p by evaluating (-1|p) = (-1)1/2(p-1). The exponent is even if p is of the form 4m + 1 and odd if p is of the form 4m + 3. It is the primitive root for only 2 and 3, so it is not considered in Artin's conjecture along with the perfect squares.

Here's a problem in Dickson's text (page 21): Show that the product of all primitive roots of a prime p > 3 is congruent to 1 mod p.

I can see that this makes sense, but I don't know how to prove it. For example, consider p = 5. The primitive roots are 2 and 3 and 2*3 = 1 mod p. One can write 3 = 23 in order to combine it with 2 and then we have 2123 = 24 which should be 1 by Fermat's theorem. So it works for one case, but how would one show that in general? That's the one I'm stuck on at the moment.

[desiresjab]

Yes, I suspect QR is a relatively recent idea.

Regarding the first wayward fact, -1 is p - 1 mod p. One can tell if -1 is a quadratic residue for odd prime p by evaluating (-1|p) = (-1)1/2(p-1). The exponent is even if p is of the form 4m + 1 and odd if p is of the form 4m + 3. It is the primitive root for only 2 and 3, so it is not considered in Artin's conjecture along with the perfect squares.

Here's a problem in Dickson's text (page 21): Show that the product of all primitive roots of a prime p > 3 is congruent to 1 mod p.

I can see that this makes sense, but I don't know how to prove it. For example, consider p = 5. The primitive roots are 2 and 3 and 2*3 = 1 mod p. One can write 3 = 23 in order to combine it with 2 and then we have 2123 = 24 which should be 1 by Fermat's theorem. So it works for one case, but how would one show that in general? That's the one I'm stuck on at the moment.

That is an interesting problem. If I may make a suggestion. My intutition is that you need to pair the primitive roots up, since there is usually an even number of them, φ(p-1) of them, actually. The trick is going to be something similar to what I did to prove Fermat's little Theorem. I believe you can pair the roots with their inverses for multiplication and get 1, since if a is a root, a-1 is also.

Notice that in your example, 2 and 3 are inverses of each other (mod 5), or else their product would not equal 1. Hope that helps a little.

[desiresjab]

In the event that φ(p-1) is not even, it should still work out. You simply multiply a times b, and that will be the inverse of the remaining c (mod p).

[desiresjab]

Sorry, duplicate post.

[desiresjab]

Now if you multiplied the entire residue system of a prime together, you should also get 1, right? For each element in the set has an inverse, which is also in the set, and there are an even number of elements in the complete reisdue system.

[desiresjab]

Come to think of it, the directly above task should actually be easier when φ(p-1) is odd, since 1 is in every set, and must be paired with itself, leaving an even number of numbers to pair.

[desiresjab]

Sorry for spreading this out, but you have made me think. The evenness or oddness of the set probably does not matter. In those cases where it seems at first likely to interfere, I would almost bet that it will magically work out because one of the numbers in the set (besides 1) will be its own inverse. I do not have hardcore evidence or a proof, but my experience is telling me that. Anyway, I think I have said enough, if not too much.

[YesNo]

Thanks, desiresjab. It makes sense to pair the primitive roots with their inverses. It didn't occur to me that a primitive root's inverse is also a primitive root, but I think that should be the case.

I found another introductory textbook on the subject by Charles Vanden Eynden which I am also reading. When I get stuck with one, I move to the other.

[desiresjab]

Thanks, desiresjab. It makes sense to pair the primitive roots with their inverses. It didn't occur to me that a primitive root's inverse is also a primitive root, but I think that should be the case.

I found another introductory textbook on the subject by Charles Vanden Eynden which I am also reading. When I get stuck with one, I move to the other.

Pairing members of sets up is a common trick in the field. Do not be afraid of pen and paper. If you can get one move closer to the nut, then you only have to see two moves deep instead of three, etc.

I love the way modular arithmetic forces even the largest numbers to play its game. It puts numbers on the rack and extracts certain truths from them. It says to the gigantic prime: you are only 2 (mod 3), pal, now get up there.

[YesNo]

I just finished the following article giving a proof of Euler's theorem using pairing based on a number and its inverse. Being relatively prime to n is the same thing as having an inverse mod n. That is (a,n) = 1 <=> there exists x such that ax = 1 mod n. I'll have to keep that in mind. http://sites.millersville.edu/bikenaga/number-theory/euler/euler.pdf

[desiresjab]

I just finished the following article giving a proof of Euler's theorem using pairing based on a number and its inverse. Being relatively prime to n is the same thing as having an inverse mod n. That is (a,n) = 1 <=> there exists x such that ax = 1 mod n. I'll have to keep that in mind. http://sites.millersville.edu/bikenaga/number-theory/euler/euler.pdf

Yes indeed. There are so many of these little facts and theorems and criteria and lemmas and adjuncts that remembering them all when one of them happens to be relevant is a matter of experience as you see more and more of the mulitiplicty of connections between ideas at the level of numbers.

Only when I can see straight through QR to the intiuitive reason it must be so, could I possibly say God could not create a universe where QR is not as it is in our universe.

The one thing absolutely necessary to our universe or any universe is the same laws of mathematics everywhere. If I can see that about QR, I can say it, but I do not see it with that particular clarity yet, and furthermore do not know if I am capable of that. It is still a goal, though.

I figure God could not create a universe where 2 is not the successor of 1. When I can see the reasons for QR as clearly as 2 succeeding 1, I will know the limits of God, at least from my human perspective.

[desiresjab]

My suspicion is, for me the key is to "see through" why two 4n+3 primes behave the way they do, which I call irreciprocity.

If I can see why they cannot behave as 4n+1 primes or as a mixed pair, I sense I can see it all by the same method.

It might be that a formal proof exists which would satisfy me fully, but I have no access to it or would not have the tools to understand it. Many proofs enter the terrain of Group theory and Abstract algebra, and depend on quite a few other proofs. I need to upgrade there, but there is not time for everything.

I do not know if it is possible to see it the way I want to see it. I am not even sure that anyone does. Reading the words of math professors on the subject over at the n-category cafe, I can see that even among that level the grasp is dubious, depending on a particular proof usually. One senses the lack of a deep intuitive connection and understanding of why the numbers must behave as they do. Usually because they belong to some group, subgroup or coset, which is shown to be symmetrical or asymmetrical, as the purpose serves, etc., but which is rather far removed from ground level.

I may be trying to see something at ground level that is not visible from ground level.

There was another book I now remember. It was by a gentleman named Weils who was something of a modern giant. This book was a treatment of numbers from a group theoretical standpoint, and I did not take it seriously enough at the time. That might be all I need, not the whole course. The thing is, I already have a decent understanding of what those fields do and say. It is simply the strange notation I have not adapted to on my own. I am such a fuss budget and whiner before I settle down and adapt to what one obviously has to do.

[YesNo]

There is a lot to understand, but I try to think of these as pieces of a jig-saw puzzle. Here are the pieces so far in my quest to solve Artin's Conjecture, at least the part that says for any number greater than 1 there are infinitely many primes for which it is a primitive root.

Puzzle Piece 1: Two integers that are relatively prime have inverses with respect to each other. In particular (a,n) = 1 if and only if there exists x such that ax=1 mod n. This means we only have to look at relatively prime integers and φ(n) would represent how many there are. If p is a prime, then φ(p) = p - 1. For simplicity stick with primes and the numbers relatively prime to them.

Puzzle Piece 2: A primitive root a multiplied by itself has to generate all the residues mod n. In particular it can't stop generating a number different from 1 until it generated all of them. Further for any d > 1 dividing p - 1, a(p-1)/d cannot equal 1 mod n. Otherwise it has stopped generating the residues and it is not a primitive root. So for d = 2, if a(p-1)/2 = 1 mod n and therefore was a quadratic residue it would not be a primitive root.

Puzzle Piece 3: If Artin's conjecture is true, then for each a > 1 there exist infinitely many primes for which a is a quadratic nonresidue. The converse is false. However, maybe this is easier to solve if it hasn't already been solved.

Puzzle Piece 4: To simplify matters, let a = 3 which is one of the 4m+3 numbers. If p is another 4m+3 prime then QR can relate them so that calculations work faster, but I can't rely on calculations since I am working with infinitely many primes p. So far QR seems good for calculation, but nothing else.

Puzzle Piece 5 (the one I'm on now): Suppose p is a 4m+3 prime and p - 1 = 2r where r is another prime. Let a = 3. If (3|p) = -1, then I have handled the case when the divisor is 2: 3(p-1)/2 = -1. Does this imply anything about the other divisor (p-1)/r? So, are there infinitely many primes where p-1 = 2r and r is prime and what additional conditions do I need to tell if 3 is a quadratic nonresidue?

At the moment I see QR's value in helping one calculate whether a number is a quadratic residue or not faster. I must be missing something important.

[desiresjab]

There is a lot to understand, but I try to think of these as pieces of a jig-saw puzzle. Here are the pieces so far in my quest to solve Artin's Conjecture, at least the part that says for any number greater than 1 there are infinitely many primes for which it is a primitive root.

Puzzle Piece 1: Two integers that are relatively prime have inverses with respect to each other. In particular (a,n) = 1 if and only if there exists x such that ax=1 mod n. This means we only have to look at relatively prime integers and φ(n) would represent how many there are. If p is a prime, then φ(p) = p - 1. For simplicity stick with primes and the numbers relatively prime to them.

Puzzle Piece 2: A primitive root a multiplied by itself has to generate all the residues mod n. In particular it can't stop generating a number different from 1 until it generated all of them. Further for any d > 1 dividing p - 1, a(p-1)/d cannot equal 1 mod n. Otherwise it has stopped generating the residues and it is not a primitive root. So for d = 2, if a(p-1)/2 = 1 mod n and therefore was a quadratic residue it would not be a primitive root.

Puzzle Piece 3: If Artin's conjecture is true, then for each a > 1 there exist infinitely many primes for which a is a quadratic nonresidue. The converse is false. However, maybe this is easier to solve if it hasn't already been solved.

Puzzle Piece 4: To simplify matters, let a = 3 which is one of the 4m+3 numbers. If p is another 4m+3 prime then QR can relate them so that calculations work faster, but I can't rely on calculations since I am working with infinitely many primes p. So far QR seems good for calculation, but nothing else.

Puzzle Piece 5 (the one I'm on now): Suppose p is a 4m+3 prime and p - 1 = 2r where r is another prime. Let a = 3. If (3|p) = -1, then I have handled the case when the divisor is 2: 3(p-1)/2 = -1. Does this imply anything about the other divisor (p-1)/r? So, are there infinitely many primes where p-1 = 2r and r is prime and what additional conditions do I need to tell if 3 is a quadratic nonresidue?

At the moment I see QR's value in helping one calculate whether a number is a quadratic residue or not faster. I must be missing something important.

Philosophically, my own inclination is toward mathematics as we know it being necessary just as it is. God could not controvert or skirt this necessity, meaning God has limitations. A limted God was an idea of John Stuart Mill.

Very, very true, I could have set the bar anywhere, I could have chosen easier propositions. But I just happened to settle on QR because I knew it was hard, I did not understand it at the time and figured I should earn the right to make such a statement as God is constrained by mathematics.

I am so close now. I again sense Eisenstein's proof as the way forward. If one cannot see it in the numbers themselves, see it in the exponents represented by those dots and X's, then backwards extrapolate to the numbers.

[desiresjab]

I think a key point to realize about Eisenstein's proof, is that in his triangles AYX and WAY, the number of lattice points in them do not reperesent actual degree of the exponent on p or q, but have the right parity, which is all that matters--odd or even. For instance, the number of even lattice points in his big traingle, with 17 even lattice points, seems to represent the true exponent on -1 for p, whereas the triangles AYX and WAY merely give the right parity for p or q, which is suffiucent, indeed, but not quite the same thing.

In the event of two 4n+3 primes, the two small triangles will have opposite parity, which forces -1 as the final outcome of the operation.

I am conjecturing that the triangles AYX and WAY will never contain the same quantity of total lattice points, but their parity will be in accord when both are not 4n+3 types.

This will at first seem strange, as the two large triangles ABC and ADC always have the same total number of lattice points, of course. This does not mean one contains the same number of even or odd points as the other, however, for indeed they do not. Only their total number is equal.

Let's forget I made that conjecture, since it is false. You could say it is usually true but not always. I think they can be equal when the two primes are near the same size and their QR value is 1, and when, of course, they are both 4n+1 types only..

[desiresjab]

This goes to show how dangerous the conjecture game in mathematics is. I amended my above post four or five times until I finally saw the truth. The quantity of lattice points in the triangles AYX and WAY can be equal when the two primes are close enough in size to each other, and at least one is a 4n+1 type. I don't think that fact even has much significance. Red Herring.

[desiresjab]

Even so, the two triangles will have different quantities of even and odd lattice points, though the total number of lattices in each is the same. I am done with that. Can those two triangles ever have the same number of even points and the same number of odd points? I don't know, y'all, and I ain't gonna think about it. However, I think they cannot. Watch out! Watch out!

[desiresjab]

Hold it, dummy (speaking to myself). The natural exponent for -1, ie., the one which duplicates (p-1)/2 (q-1)/2, is found by summing the total number of lattice points in the triangles AXY and WAY. Ah, now that is good. We have gotten somewhere.

In the case of 11 and 13 for p and q, the exponent would be 30, which is even and therefore produces 1 when used as an exponent.

-1(11-1)/2 (13-1)/2=-130=1

[YesNo]

Here are some links that I plan to look at more closely on Eisenstein's proof of QR to see if I can understand this. Do you have some links?

https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity

http://math.ucsb.edu/~jcs/QuadraticReciprocity.pdf

[desiresjab]

Here are some links that I plan to look at more closely on Eisenstein's proof of QR to see if I can understand this. Do you have some links?

https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity

http://math.ucsb.edu/~jcs/QuadraticReciprocity.pdf

I have used so many cites I could not begin to dig them up.

Make a p X q rectangle on graphing paper. Draw a diagonal carefully. 19 by 23 was the largest rectangle my paper allowed me to draw. There are enough primes below 23 to get the picture.

The exact number of lattice points corresponding to (p-1)/2 (q-1)/2 will be found in the triangles AYX and WAY within the rectangle (p/2)(q/2). Watch what happens in those two triangles as you construct rectangles for different primes and prime types. The total number of points in either triangle is the same, they do not have the same quantity of odds or evens.

The borders of (p/2)(q/2) are between lines on the graphing paper, ensuring that we have no lattice points on the perimeter. The perimeter of the invisibleble rectangle with lattice points on the perimeter of it would have dimensions (p-1)/2 (q-1)/2. This is the geometric connection between (p-1)/2 (q-1)/2 and (p/2)(q/2). I am not sure how clear that is. I am trying to bring you up to my current understanding on the problem.

I see it now. I see how those small triangles work and why they do. I am now about 98% satisfied with my undertstanding of Eisenstein's proof. It is no longer a mystery why the lattice points and the exponents match up.

God could only create a universe where QR is true, if our imaginations are asked to judge. No QR in our universe is every bit as absurd a notion to our brains as 2 is not the successor of 1. Over and out.

[YesNo]

I see the diagram. Also it makes sense that there is no lattice point on the diagonal line, y = (p/q)x, since p and q are primes. That is, an integer value for x would not make y an integer. I also see how there are (p-1)/2 (q-1)/2 lattice points in the two triangles, AYX and WAY. What I don't see is the connection between those lattice points and something that will discriminate between 1 or -1. All I can see is the overall count is correct. This seems to me like I am missing something.

I remember reading that Galileo pointed his telescope to Jupiter and asked one of his friends to look. Even though his friend was willing to agree with him, he didn't understand that what he was looking at were Jupiter's moons rather than more stars and so the evidence didn't convince him. I am sort of like that with this proof at the moment. Proofs are like spaghetti code until one understands them. Unraveling the spaghetti takes time. After understanding, one can try making a new proof that might be easier to understand. I hear there are hundreds of proofs for QR.

The Gauss Lemma makes more geometric sense to me at the moment than this one does. Start with Fermat's Theorem, ap-1 = 1 mod p. The quadratic residues would have a(p-1)/2 = 1 mod p as well. And so one already has one way to calculate whether a is a quadratic residue or not by repeated multiplication of a. The lemma replaces a with -1, which simplifies the calculation as far as the multiplication goes, but also complicates it since the exponent is no longer (p-1)/2, but the number of negative elements when reduced mod p to numbers between -(p-1)/2 and (p-1)/2. Finding that exponent is now the hard part.

Geometrically that makes sense to me and, by the way, it also helps solve the puzzle piece I was working on. It seems there are infinitely many primes p for which 3 is a quadratic nonresidue and hence a potential primitive root. I would now need to generalize that.

[desiresjab]

Sometimes it is difficult for me to determine if you are talking about QR or PR, or searching for a connection between the two. I have to go out on a limb and say provisionally I do not think the two are hugely connected. They are connected some way, however, because just about all number theoretic functions are connected, no matter how distantly. Their connection may even be important.

This could very well be a fault in my own vision. From looking at the problem so long my own way I may have developed myopia. My brain is open for business, though.

I know for a fact that Eisenstein went into his proof with much knowledge. He already knew the significance of (p-1)/2 and (q-1)/2, which is why he made the lattice points in AXY and WAY match up to them one-to-one.

At this point we have much knowledge, too. For instance, we do not have to consult anything to know that a 4n+1 prime will always have -1 as a quadratic residue and 4n+3 primes will never.

You seem to be aking: where is this information located in Eisenstein's rectangles? I have to say at this point I do not know if it even is. This is information we already know, and I am unaware of that information being graphically represented in Eisenstein's diagram at this point. I will search for its presence, however, for I am not fool enough to think there is nothing obvious I might be missing.

[desiresjab]

Gauss knew Eisenstein. He may have beeb Gauss's student. What I suspect is that Eisenstein found the most elementary proof possible. This what mathematicians always strive for. If one man's proof requires calculus and a second man's proof requires only algebra, the second proof is considered more elegant.

For Gauss to have been all around this proof only to have Eisenstein find and present it--did this rasp the old man? Gauss demoted Euler because he was so close to QR and did not get it. Yet he stood within inches (figuratively) of this be-all end-all of QR proofs.

Hundreds more proofs were to come, but we know none are as elegant as Eisenstein's. The fact that Wikipejia chose it is testament of this. Every other proof I have looked at is a devil, and requires higher concepts.

What was Gauss thinking when he made his famous comment about Eisenstein? Gauss made seven or eight proofs of QR in his lifetime. I would be willing to bet each illuminated a different aspect of it, or Gauss would not have bothered. The fact that he was still working on it throughout his lifetime probably means even he, the mightiest of mighty, felt he did not have full grasp of it. Why else would a man with so many other important things to get to still be fussing with QR decades after he solved it?

This means we sure as heck do not have to feel bad or guilty for only having partial understanding of this theory. Gauss had the telegraph to invent, conformal mapping to forrmalize, magnetism to overhaul, differential geometry to launch, yet he kept coming back to QR his entire life to produce more proofs. Ask youself, would he have done this if he had every bit of understanding he felt he needed on the topic?

He felt it was his crowning acheivement. This fellow who as a teenager cracked a seventeen hundred year old problem that had puzzled the ancients, who formalized modular arithmetic, who presented the first proof of the fundamental theorem of algebra, who built the algebraic structure for imaginary numbers--he considered QR the greatest (and perhaps the deepest) of his acheivements.

If we understood QR quickly and easily, something would be wrong. Powerhouses like Gauss and Euler and Legendre do not struggle mightily only for us to come along and breezily understand at will. Make no mistake about it, this stuff is hard. QR is a gateway to the really hard in number theory. It is always presented at the end of elementary number theory couses. After that you are no longer on the wading end of the pool--you swim or sink in those deep waters.

[Dreamwoven]

I wonder if this website would be helpful?
http://math.stackexchange.com/questions/43579/qr-with-column-pivoting

[YesNo]

I looked up "QR decomposition" and it seems to be concerned with factoring matrices. https://en.wikipedia.org/wiki/QR_decomposition It may be related, but I don't see how at the moment. Quadratic reciprocity, which we abbreviated here as QR, is about whether an integer x is a square modulo a prime p, that is, does there exist an integer r such that r2 = x mod p? If so (x|p), the Legendre notation for whether x is a quadratic residue mod p, would equal 1.

What I am looking at is the Artin's conjecture which says given a number, m>1, there are infinitely many primes for which m is a primitive root. That is, multiply m by itself over and over again and all the elements of the reduced residue system mod that prime are generated. If m is a primitive root then it is also a quadratic non-residue, otherwise it would not generate all the elements, but stop half way through.

Desiresjab is interested in quadratic reciprocity and in particular Eisenstein's proof of it. I find that interesting also, because the more I learn about that the more I understand why Artin's conjecture is hard to solve.

Here is an outline of a proof of quadratic reciprocity using Eisenstein's lattice points: http://math.ucr.edu/home/baez/136/quadratic.pdf

The article doesn't prove anything. It just states the propositions, which is frustrating, but it only claimed to offer a "big picture". The proposition that gets me stuck is called in that paper "Baby Eisenstein's Lemma". It says that the number of points in the lattice in the lower triangle in Eisenstein's drawing has the same parity (even or odd) as the number of elements in Gauss Lemma that fall in the negative part of the reduced residue system from -(p-1)/2 to (p-1)/2). If we know how many there are then m is a quadratic residue if that number is even and a quadratic non-residue if that number is odd. So, it is not that they match one-to-one but that they have the same parity.

That's the clue I am following at the moment. It is only a parity issue between those lattice points and the elements in Gauss Lemma. However, I don't know how to prove that which means I don't understand it.

[desiresjab]

I looked up "QR decomposition" and it seems to be concerned with factoring matrices. https://en.wikipedia.org/wiki/QR_decomposition It may be related, but I don't see how at the moment. Quadratic reciprocity, which we abbreviated here as QR, is about whether an integer x is a square modulo a prime p, that is, does there exist an integer r such that r2 = x mod p? If so (x|p), the Legendre notation for whether x is a quadratic residue mod p, would equal 1.

What I am looking at is the Artin's conjecture which says given a number, m>1, there are infinitely many primes for which m is a primitive root. That is, multiply m by itself over and over again and all the elements of the reduced residue system mod that prime are generated. If m is a primitive root then it is also a quadratic non-residue, otherwise it would not generate all the elements, but stop half way through.

Desiresjab is interested in quadratic reciprocity and in particular Eisenstein's proof of it. I find that interesting also, because the more I learn about that the more I understand why Artin's conjecture is hard to solve.

Here is an outline of a proof of quadratic reciprocity using Eisenstein's lattice points: http://math.ucr.edu/home/baez/136/quadratic.pdf

The article doesn't prove anything. It just states the propositions, which is frustrating, but it only claimed to offer a "big picture". The proposition that gets me stuck is called in that paper "Baby Eisenstein's Lemma". It says that the number of points in the lattice in the lower triangle in Eisenstein's drawing has the same parity (even or odd) as the number of elements in Gauss Lemma that fall in the negative part of the reduced residue system from -(p-1)/2 to (p-1)/2). If we know how many there are then m is a quadratic residue if that number is even and a quadratic non-residue if that number is odd. So, it is not that they match one-to-one but that they have the same parity.

That's the clue I am following at the moment. It is only a parity issue between those lattice points and the elements in Gauss Lemma. However, I don't know how to prove that which means I don't understand it.

The number of lattice points in Eisenstein's triangles AYX and WAY give the exact value of the exponents, not just the correct parity.

[desiresjab]

That is, they give the correct sum of total exponents. (12X10)/4=15+15

[desiresjab]

As far as I can tell quadratic reciprocity and QR as used in those matrices are different functions. I think QR means something else with regard to those matrices. I do not think it means quadratic reciprocity.

[desiresjab]

My posts are disappearing.

[desiresjab]

My own unsloved problem is Brocard's problem. In mod arithmetic it might be stated thus:

q2≡1 (mod (p!)). Things that look as if they should be simple, turn out to be near impossible.

[YesNo]

The number of lattice points in Eisenstein's triangles AYX and WAY give the exact value of the exponents, not just the correct parity.

That's true. The exponent (p-1)/2*(q-1)/2 are the number of lattice points within the triangles AYX and WAY. However, the Gauss Lemma comes up with a smaller exponent. My difficulty is how to show that the smaller exponent can be replaced by the larger one so that the calculation depends only on p and q.

For example, let p = 11 and q = 13. Then (p-1)/2 = 5 and (q-1)/2 = 6. To see what the Gauss Lemma provides consider the numbers from 1 to (13-1)/2 = 6 or {1,2,3,4,5,6} and multiply them by p = 11 mod 13. This gives {11,9,7,5,3,2}. Values over 6 could be viewed as negative if we used the residue set between -6 and 6 mod 13 rather than the one between 0 and 12 mod 13. There are 3 values larger than 6, namely, {11,9,7}, and -13 = -1 = 11(13-1)/2 mod 13 = 116 mod 13. That would be (p|q) = (11|13) = -1. The Gauss Lemma states that is another way to calculate (p|q) rather than to raise p to the (q-1)/2 power mod q.

Considering (13|11), we would look at this set of residues mod 11 {1,2,3,4,5}. By the Gauss Lemma we multiply each of them by 13 and get the following {2,4,6,8,10}. Now we count those in the set greater than 5 = (11-1)/2 and find there are 3 of them. So -13 = -1 = 13(11-1)/2 mod 11 = 135 mod 11. That would be (q|p) = (13|11) = -1.

Since p = 11 = 3 mod 4 and q = 13 = 1 mod 4, quadratic reciprocity says that (p|q) = (q|p) which turns out to be the case since they both equal -1.

My problem is, I understand the proof of Gauss Lemma which gives exponents of 3 for both p and q. But I don't see how either Gauss or Eisenstein raises that exponent to 5 and 6. I can see why they would want such exponents. It would would be easier to calculate.

[desiresjab]

That's true. The exponent (p-1)/2*(q-1)/2 are the number of lattice points within the triangles AYX and WAY. However, the Gauss Lemma comes up with a smaller exponent. My difficulty is how to show that the smaller exponent can be replaced by the larger one so that the calculation depends only on p and q.

For example, let p = 11 and q = 13. Then (p-1)/2 = 5 and (q-1)/2 = 6. To see what the Gauss Lemma provides consider the numbers from 1 to (13-1)/2 = 6 or {1,2,3,4,5,6} and multiply them by p = 11 mod 13. This gives {11,9,7,5,3,2}. Values over 6 could be viewed as negative if we used the residue set between -6 and 6 mod 13 rather than the one between 0 and 12 mod 13. There are 3 values larger than 6, namely, {11,9,7}, and -13 = -1 = 11(13-1)/2 mod 13 = 116 mod 13. That would be (p|q) = (11|13) = -1. The Gauss Lemma states that is another way to calculate (p|q) rather than to raise p to the (q-1)/2 power mod q.

Considering (13|11), we would look at this set of residues mod 11 {1,2,3,4,5}. By the Gauss Lemma we multiply each of them by 13 and get the following {2,4,6,8,10}. Now we count those in the set greater than 5 = (11-1)/2 and find there are 3 of them. So -13 = -1 = 13(11-1)/2 mod 11 = 135 mod 11. That would be (q|p) = (13|11) = -1.

Since p = 11 = 3 mod 4 and q = 13 = 1 mod 4, quadratic reciprocity says that (p|q) = (q|p) which turns out to be the case since they both equal -1.

My problem is, I understand the proof of Gauss Lemma which gives exponents of 3 for both p and q. But I don't see how either Gauss or Eisenstein raises that exponent to 5 and 6. I can see why they would want such exponents. It would would be easier to calculate.

5 and 6 are the effective dimensions of the smaller rectangle comprised of the triangles AYX and WAY. These match up with Euler's criterion. These are the ones you want, I believe. Adding 1/2 to their dimensions allows lattices points on the perimeter of the smaller rectangle with dimension 5 and 6. Therefore they are all in the interior of the rectangle augmented by 1/2 in its dimensions.

5 and 6 get the right product, but I do not see anything in the triangles denoting a significance of 5 and 6, other than their product and their opposite parity. I also see nothing which tells me about -1, other than a fact we already know--that -1 is a residue of all 4n+1 primes. Since we know that already about primes, it does not seem important to me that the diagram does not speak to that aspect.

[YesNo]

It looks like the exponent of -1 in the following


(p|q) = (-1)(p-1)(q-1)/4 (q|p)

is just an algebraic way to write the English phrase


(p|q) = (q|p) unless p and q are both congruent to 3 mod 4 in which case (p|q) = -(q|p).

It also occurred to me the main reason for quadratic reciprocity is for calculation purposes. This theorem allows us to flip the p and q and then reduce the larger one.

[desiresjab]

It looks like the exponent of -1 in the following


(p|q) = (-1)(p-1)(q-1)/4 (q|p)

is just an algebraic way to write the English phrase


(p|q) = (q|p) unless p and q are both congruent to 3 mod 4 in which case (p|q) = -(q|p).

It also occurred to me the main reason for quadratic reciprocity is for calculation purposes. This theorem allows us to flip the p and q and then reduce the larger one.

I'm not too sharp right now. I have come down with the flu or something. Reciprocity is easy to state in English. The first step to understanding it is to learn enough math (mostly modular) to understand the English.

Now of course it has a much bigger reason--the razzle dazzle of ciphers.

I'll be back after I throw up and sleep.

[YesNo]

Get well!

The more I look at this the more I feel like I am still in the shallow end of the pool.

Edit: After reading a proof, different from Eisenstein's, I began to see how the primes p and q are connected: They are each related to their product, pq.

Also the proof depended upon something nearly obvious: if three integers are added together and their sum is an even number then either all three of the integers are even or only one of them is.

http://www.lehigh.edu/~shw2/q-recip/gauss5.pdf

[desiresjab]

Get well!

The more I look at this the more I feel like I am still in the shallow end of the pool.

Edit: After reading a proof, different from Eisenstein's, I began to see how the primes p and q are connected: They are each related to their product, pq.

Also the proof depended upon something nearly obvious: if three integers are added together and their sum is an even number then either all three of the integers are even or only one of them is.

http://www.lehigh.edu/~shw2/q-recip/gauss5.pdf

Back with better health.

How Gauss determines that what me might call the overflow values are enough to determine the quadratic relationship is algebraic magic, of course, but not necessarily transparent. It gives another fact, but the torch in the fact is hard to light. There is extreme familiarity with modular operations and the Chinese remainder theorem and how they all apply. Seeing how every rule you need applies at the right time and place is always going to be the case. In one's personal investigations, if one misses one of these, a great deal of time can be lost but not necessarily wasted in chasing down proof along the way of something that boils down after all to a basic law of modular arithemtic the explorer has not yet assimilated into his mathematical vocabulary fully enough so that its applications and assumptions come naturally as they do in normal algebra.

No matter how simple they continue to try to make QR, it always turns out pretty complex, except in terms of the laws themselves, which are clear and easy to apply. In asking why the two species of primes themselves behave the way they do with themselves and with the other species, it is profitable to remember that they only do so in modular arithmetic, where QR is a theorem. The comparison to normal arithmetic in the Martinson link I gave earlier is still the most illuminative and suggestive article I have seen yet. For that reason, I believe a good review of Fermat's and LaGrange's sums of squares is in order. I did this in cursory fashion a few months back, without settling in for the full ride with different hosts.

Another nagging propositon I looked at only a few weeks ago is Bertrand's paradox. It deals with geometry and the power of the continuum infinity of points on the surface of a sphere. Someone gives a good explanation on YouTube of how it is mathematically possible to dissect a sphere of diamter X and reassemble the parts into two full and complete spheres of diameter X without adding any new material. No man or machine could actually make these slices and chops, but in theory it is feasible. Or is it an unresolved paradox of infinite set theory? It deserves a second dip. So many things to chase down.

[desiresjab]

To continue in the same vein, the quote But this is just the set of integers, lifted directly from the end of the Gauss proof is, in English, what always happens at the conclusion of proofs in this mode of math, now isn't it? The results of operations on a residue system are shown to be equivalent to another set of integers previously defined. When all you need is parity to prove your point the sets do not even need the same cardinality to yield their information. This is heading toward set theory and group theory. I am still stuck on the idea of an easier vantage to peer at the heart of the law and see naked numbers bathing.

[desiresjab]

Deciding how one can show some particular set equals another set or subset must be the normal way to proceed then, what you strive to frame your question in terms of in modular forests. That is my nutshell observation.

[YesNo]

Another nagging propositon I looked at only a few weeks ago is Bertrand's paradox. It deals with geometry and the power of the continuum infinity of points on the surface of a sphere. Someone gives a good explanation on YouTube of how it is mathematically possible to dissect a sphere of diamter X and reassemble the parts into two full and complete spheres of diameter X without adding any new material. No man or machine could actually make these slices and chops, but in theory it is feasible. Or is it an unresolved paradox of infinite set theory? It deserves a second dip. So many things to chase down.

I looked around on YouTube and found this description of Bertrand's Paradox: https://www.youtube.com/watch?v=uI2FnUmBeeo

It seems that the paradox is resolved once one defines what it means to "choose a chord at random". One of the choices started with a fixed point, another with a fixed diameter the chords had to cross and the third asked whether the midpoint of the chord was inside or outside an interior circle. Not all of the possible chords were permitted by the selection constraints in the first two examples. I suspect the third example did include all possible chords.

[YesNo]

Bertrand's problem looks harder than I realized: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

I don't think the problem is resolved, as I claimed above, by saying there are more chords in one of the three examples. One can assume there is one chord in each. Then what is the probability that its side is longer than the side of an inscribed equilateral triangle. I think you are right in looking at this as a problem of picking a point from an infinite number of points.

[stavrost]

my recollection of time is that it began with the universe, (space-time). so the first thing we have to do is wrap our heads around the question: if space-time began with the universe, then how can we ask the question, "what was before?" before implies time, but it didn't exist.

[stavrost]

A fascinating idea that i have wondered about is the Schrodinger's cat conundrum. if there has to be an observer before any event can take place, then there had to be an observer before the first two particles o matter inter-acted, did there not? This has led some to theorize that there was intelligence prior to matter, rather than the other way around.

[YesNo]

The Schrodinger cat problem keeps bothering me as well. Basically, every time I think I understand what it is supposed to show, I doubt that I have it right. Although I haven't read much lately, I have Amit Goswami's "The Self-Aware Universe" on my desk. He promotes "idealist science" as opposed to "materialist science".

My current view is slightly different from "there was intelligence prior to matter". At the moment, I don't think unconscious matter exists. There is nothing but intelligence. What we see as matter is conscious at a lower level that appears unconscious at the macro level where we view it.

Of course, I might be completely wrong.

[tailor STATELY]

Let me know if this is the wrong place...

"The superfluid Universe":
We are used to thinking that quantum physics dominates only the microscopic realm. But the more physicists have learned about quantum theory, the more it has become clear that this isn’t so. Bose-Einstein condensates are one of the best-studied substances that allow quantum effects to spread widely through a medium. In theory, quantum behaviour can span arbitrarily large distances, provided it isn’t disturbed too much. https://aeon.co/essays/is-dark-matter-subatomic-particles-a-superfluid-or-both?utm_medium=email&utm_source=digg

... and a toon: http://www.gocomics.com/bloom-county

Ta ! (short for tarradiddle),
tailor STATELY

[Dreamwoven]

I'm on this list and though I don't understand all the mathematical stuff the main players here write, I enjoy the discussions between desiresjab and YesNo. Read the thread and you will see what I mean.

[desiresjab]

I was offline for half a week with computer issues. The only thing to do was to go to pen and paper.

What do we know about squares in general? What is one fact of odd squares?

[YesNo]

What do we know about squares in general? What is one fact of odd squares?

I don't know except the obvious in the integers, Z, that one gets an odd number when an odd number is squared and and even number when an even number is squared.

[YesNo]

Let me know if this is the wrong place...

"The superfluid Universe": https://aeon.co/essays/is-dark-matter-subatomic-particles-a-superfluid-or-both?utm_medium=email&utm_source=digg

... and a toon: http://www.gocomics.com/bloom-county

Ta ! (short for tarradiddle),
tailor STATELY

I liked the cartoon.

I didn't know that the theories around dark matter had two major variations, those promoting modified gravity and those promoting a cold particle. The superfluid idea is also new to me. Maybe it can bridge the ideas. The modified gravity reminds me of a talk by Rupert Sheldrake where he questioned whether physical constants, in particular G, were actually constant, but changed. If G changed that would be one way to get modified gravity.

[desiresjab]

I don't know except the obvious in the integers, Z, that one gets an odd number when an odd number is squared and and even number when an even number is squared.

I thought a very simple fact might be surprising. All odd squares are 4n+1 numbers. There are no 4n+3 squares; such an animal cannot exist.

[desiresjab]

An important fact for anyone trying to learn modular arithmetic has to do with symmetry. In normal arithmetic the negative and positive integers have symmetry across the point zero on the number line. That is, the absolute value of -5, for instance, is equal to 5. In modular arithmetic with primes, this is no longer true--

-5≡6 Mod 11.

Perfect multiples of the modulus have familiar symmetry across zero, but no other residue class does.

Under any prime modulus p, start squaring the positive integers n in succession. The series will always begin with the standard squares you are familiar with...1,4,9,16..., until n becomes greater than p1/2 (the square root of p), at which point n2 will wrap around the modulus to some value.

Quadratic reciprocity is about how two moduli wrap around each other under quadratic pressure.

[desiresjab]

We may further notice that when both p and q are primes of the species 4n+3, their squares always wrap around the other modulus so that p2≡2 (mod q) and q2≡1 (mod p), or vice versa. They will always express this relationship when squared within the modulus of the other. This is a fact of the universe, as inviolate as 2 is the successor of 1.

One always checks first to see if the larger of p and q simply reduces to a familiar square under the other as modulus. If so, the work is done. Otherwise one starts squaring n's to see if any wraps around to the value of p, under q as modulus, or vice versa.

[desiresjab]

Not surprisingly, then, two 4n+3 primes p and q wrap around each other just as their squares do. Either 4p+3≡1 (mod 4q+3) and 4q+3≡2 (mod 4p+3), or vice versa.

We must remember that if a≡b (mod m), then ap≡bp (mod m).

[desiresjab]

It appears my recent posts on 4n+3 only apply if one of the 4n+3 primes is 3 itself. More later. Three is not typical. Or is it?

[YesNo]

I thought a very simple fact might be surprising. All odd squares are 4n+1 numbers. There are no 4n+3 squares; such an animal cannot exist.

That is a more interesting fact than the one I presented. Here's my proof of it since it is not immediately obvious:

Assume there exists an odd square m2 congruent to 3 mod 4 to get a contradiction. There are two cases to consider: m is either congruent to 1 mod 4 or m is congruent to 3 mod 4.

Consider the first case, m ≡ 1 (mod 4). Then there exists r such that m = 4r + 1 and m2 = (4r + 1)(4r + 1) = 16r2 + 8r + 1 which is congruent to 1. So m is not congruent to 1.

Consider the second case m ≡ 3 (mod 4). Then there exists s such that m = 4s + 3 and m2 = (4s + 3)(4s + 3) = 16s2 + 24s + 9. Since 9 is congruent to 1 mod 4, m is not congruent to 3.

In all cases m is not congruent to 3 mod 4 and since this contradicts the assumption, the assumption is false.

[YesNo]

We may further notice that when both p and q are primes of the species 4n+3, their squares always wrap around the other modulus so that p2≡2 (mod q) and q2≡1 (mod p), or vice versa. They will always express this relationship when squared within the modulus of the other. This is a fact of the universe, as inviolate as 2 is the successor of 1.

One always checks first to see if the larger of p and q simply reduces to a familiar square under the other as modulus. If so, the work is done. Otherwise one starts squaring n's to see if any wraps around to the value of p, under q as modulus, or vice versa.

I would say it is a fact of the axiom system and the set of elements one is using rather than a fact of the universe. One could change the axiom system or the set of elements and get something different. For example, Euclidean geometry need not have much to do with space in the universe around us, but the results would be inviolate facts within the axioms of Euclidean geometry. Only if one can't consistently change the axioms would it be possible to look at the results as relevant to the universe.

I agree that the computationally hard part comes from the wrapping process.

[desiresjab]

I would say it is a fact of the axiom system and the set of elements one is using rather than a fact of the universe. One could change the axiom system or the set of elements and get something different. For example, Euclidean geometry need not have much to do with space in the universe around us, but the results would be inviolate facts within the axioms of Euclidean geometry. Only if one can't consistently change the axioms would it be possible to look at the results as relevant to the universe.

I agree that the computationally hard part comes from the wrapping process.

You can do these things anywhere. Euclidian geometry would be an outside geometry in some universes. Its laws would remain true, just as the laws of non-Euclidian geometries are true for us.

The wrapping process of moduli can turn a 4n+1 square into a 4n+3 number since, for instance three is a square under some moduli.

I only need to pinpoint the mechanics that force (4n+3)2 to perform its consistent behavior under prime moduli, for the whole thing to shake out. It is a matter of mechanics. A mechanical detail is eluding me so far. That detail will clear up every question. I not only sense this is true, I know damned well it is. There is no doubt, either, that that detail is clearly available in group theory, which is why so many proofs rely on it.

I still believe it is something I can get from peering at the numbers. My investigations are going deeper underground where I need paper and pencil.

Remember the Martinson list for the sums of squares? He mentioned that any prime number generated by a sum of two squares was a 4n+1 number. He mentioned that the table would generate every prime of 4n+1 makeup. He did not mention that all odd numbers in the table were also 4n+1 numbers. Close inspection reveals that sums of two squares can only be 4n, 4n+2, or 4n+1 numbers.

On another note of interest. Breaking a large 4n+1 prime into its unique sum of two squares, is every bit as difficult as factoring. I have not delved deeply enough, but I wonder if any of the present encryption systems are taking advantage of this. A new function as the basis means no patent battles.

Anyway, I feel I am very close to the final solution with QR. I know where to dig and I think I know how to do it.

[desiresjab]

Here is a curious fact about 4n+3 primes. Look at seven and its squares with regard to other 4n+3 primes.

Because of what we already know, we can state unequivocally that no 4n+3 prime greater than 49 can ever wrap back to be a square (mod 7). Why? Because 72 is a normally occuring number (mod 59) and it will be the square between the two, since there can always and only be one square between 4n+3 primes.

72 is the naturally occuring quadtratic residue of every 4n+3 prime larger than 47, not the other way around, ever.

This idea has its way of working with 4n+1 primes and mixed couples, too. If the larger prime does not reduce back to a sqaure under the smaller prime, then the smaller one will not stretch to a sqaure either, by the rules.

[desiresjab]

Therefore, under any moduli if p2<q, p2 is always a natural residue, meaning there is no wrap around by the sqaure.

Among 4n+1 primes and mixed couples, this fact forces them to both be squares, and to never not be mutual squares, since they must act the same way as each other. There is only a question whenever q is < p2, or vice versa. Otherwise, the results are automatic. Of course, we still must explain why they behave the way they do when p2<q. It is always good to see the task more clearly.

[desiresjab]

And one further miscellaneous fact. All square n= the sum of the nth and (n-1)th triangular numbers.

[YesNo]

And one further miscellaneous fact. All square n= the sum of the nth and (n-1)th triangular numbers.

This fact makes sense. Geometrically the two triangular numbers are on each side of a diagonal through a square matrix of points missing any lattice point. One triangular set of points will have n lattice points on a side and the other triangular set of points will have n-1 lattice points.

[desiresjab]

I state so many things incorrectly on my way to getting them right, that I should not be writing here on the subject of QR until I finish with it.

I think I am going to find that understanding why it works will prove a lot more difficult than understanding the mere mechanics of it. Its outside mechanics are not so bad, but what makes its guts work is a lot tougher to see.

[YesNo]

One reason to continue writing is to help clarify it for yourself. I don't mind reading it. It gives me something to think about. If it wasn't for you I wouldn't be thinking about any of this.

I looked at Eisenstein's proof on Wikipedia: https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity

Pieces of it are starting to click. I don't understand Eisenstein's lemma yet, but I think I see the geometric point. The goal is to show that the number of lattice points within AXYW has the same even or oddness (in the same residue class mod 2) as the points in ABC with the x coordinate an even number.

The reason the even x coordinates are needed is because he will consider the lattice points in XBCY and note that the lattice points in XBCZ are even because q-1 is an even number, each column has an even number of lattice points. Since XBCY and YCZ partition an even number of lattice points into two sets, the two sets have the same even or oddness. Then one can flip XCZ onto AXY. Now we already have the lattice points under even x coordinates. This flipping gives us the lattice points under odd x coordinates. So we have them all. Do this for AYW and we have all the lattice points, all (p-1)(q-1)/4 of them, in AXYW which is what we wanted.

The only piece missing is Eisenstein's lemma which is in the article, but I haven't finished understanding it yet.

Edit: I think I understand the lemma. It is interesting how it also uses the fact that when an even number is represented as the sum of two other integers, those two other integers are both either even or both odd. It cannot be the case that one is even and the other odd because their sum is even.

[desiresjab]

Yes, identical parity is a matter of (mod 2) with those columns. This is one of the first things Eisenstein makes clear.

P vs Q, when both are 4n+3 species. If the larger p wraps down to a square (mod q), then numbers under the smaller q may not square and then wrap to a square number (mod p).

11 (mod 7) wraps down to 4, a square (mod 7). Remember, there are only three different squares (mod 7), 1, 2 and 4. Any number squared (mod 7) wraps down to one of those three numbers.

Conversly, since 7 is >the square root of 11, then 4, 5, 6 7, 8, 9 or 10 squared would have to wrap back to 7, but something prevents them. I am now looking for exactly what prevents them from having reciprocity instead of irreciprocity. My goal is to see why two 4n+3 primes behave toward each other as they do. What prevents them from behaving as two 4n+1 primes or a mixed couple do? I feel I am on the right vein, looking for the exact spot to sink my pickaxe.

Your mention a few posts back that Gauss comes up with smaller numbers, but with correct parity, of course, was a good shout out.

It is not square vs square, in my current vision, but square vs some multiple of the other prime, which determines the precise behavior. This information is probably not in Eisenstein's rectangle, but I will not say for sure. There may be some reflection of it, but I think not as well, since there is a limit to the information the rectangle can contain.

[YesNo]

If one looked at Eisenstein's lattice points in the rectangle AXYW, the only time that rectangle will have an odd number of lattice points, as it does in the p = 7 and q = 11 example, is when both p and q are congruent to 3 mod 4. Otherwise AXYW will have an even number of lattice points. Also the number of lattice points in AXYW is equal to the sum of the number of lattice points in the triangle AXY and the number of lattice points in the triangle AYW.

If we have an odd integer that is the sum of two other integers, then one and only one of those other integers can be odd, that is, one and only one of (p|q) and (q|p) can equal -1 and be a quadratic non-residue. The other has to be a quadratic residue.

[desiresjab]

If one looked at Eisenstein's lattice points in the rectangle AXYW, the only time that rectangle will have an odd number of lattice points, as it does in the p = 7 and q = 11 example, is when both p and q are congruent to 3 mod 4. Otherwise AXYW will have an even number of lattice points. Also the number of lattice points in AXYW is equal to the sum of the number of lattice points in the triangle AXY and the number of lattice points in the triangle AYW.

If we have an odd integer that is the sum of two other integers, then one and only one of those other integers can be odd, that is, one and only one of (p|q) and (q|p) can equal -1 and be a quadratic non-residue. The other has to be a quadratic residue.

This all tue, and these are the same things I keep repeating to myself. But I know there is a mechanism in back of it all which prevents two 4n+3 primes from having the same "character." I am still in search of the precise mechanism and now have only decent confidence that I will succeed.

There is a shortcut I am not sure you know. Suppose we have a Legendre symbol (p/q), and q is much larger. Instead of squaring numbers (mod q) to see if one is a square, we are to always permitted to invert the symbol to (q/p). Then all we have to do is reduce. We only need one prime's quadratic character to know the other's, and this works equally with both species. One hundred percent legal.

Another legal move takes advantage of the symbol's mulipicative properties. if p=a(b), then (a/q)(b/q) will always give the right answer.

[YesNo]

There seems to be more going on here as you mention.

I am aware of the complete multiplicative nature of the Legendre symbol and that one can reduce the larger prime mod the smaller one. That reduced number will not likely be a prime so one would have to factor it. Also (a2|p)=1, so these even powers can be discarded.

I am looking at Vanden Eynden's "Number Theory". He proves QR using Gauss's methods. I'll see if I can find something more enlightening by using that proof.

[desiresjab]

There seems to be more going on here as you mention.

I am aware of the complete multiplicative nature of the Legendre symbol and that one can reduce the larger prime mod the smaller one. That reduced number will not likely be a prime so one would have to factor it. Also (a2|p)=1, so these even powers can be discarded.

I am looking at Vanden Eynden's "Number Theory". He proves QR using Gauss's methods. I'll see if I can find something more enlightening by using that proof.

By reducing, I only mean this: wrap the bigger number around the smaller modulus until the remainder is revealed, the way one can reduce 31 to 1 (mod 3). In our manipulations, it is always legal to invert the Legendre symbol, then reduce, since it is generally easier to "reduce" than to start squaring numbers to see if a square appears under the other modulus. I hope that was clear.

What I am looking for is probably not available, for I have never seen it mentioned that someone was explaining the mechanism of QR, they were only proving it. Gauss proved the law itself, and many others have since, but no one has explained it that I know of. If this is pulled off on a literature forum, it will be a coup for the ages.

[YesNo]

We might as well try explaining it to each other. Whether it is quantum physics or number theory, it probably doesn't make complete sense even to the people who know what they're talking about.

I think I understood what you meant by reducing the top prime in the Legendre symbol, (p/q). One might as well start with p > q and then find what p is congruent to modulo q. That won't likely be a prime any more, but one can try factoring it to simplify the calculation even more.

After looking at Vanden Eynden's text, I found this relationship which might be interesting and is part of his (Gauss's) proof of QR.

Let p and q be odd primes with p > q and p is congruent to q mod 4. Then there exists some integer a such that p = q + 4a. Now that last equation implies the existence of three other congruence relationships.

1) If p = q + 4a, then p ≡ q (mod 4a). This is just the original congruence including a.
2) If p = q + 4a, then p = 4a + q and so p ≡ 4a (mod q). Now, 4a is linked to p via q.
3) If p = q + 4a, then -q = 4a + p(-1) and so -q ≡ 4a (mod p). Now 4a is linked to -q mod p.

In this way 4a is the link between p and q. One can get the QR result for the cases when p ≡ q (mod 4) by considering the following:

(p/q) = (4a/q) = (4a/p) = (-q/p) = (-1/p)(q/p)

The first part comes from p ≡ 4a (mod q). The second part was proved in the book and is non trivial, but can be assumed for the moment. The third part comes from 4a ≡ -q (mod p) and the last part comes from the multiplicative property of the Legendre symbol.

I'm not sure if this helps any, but it seemed interesting to me.

[desiresjab]

We might as well try explaining it to each other. Whether it is quantum physics or number theory, it probably doesn't make complete sense even to the people who know what they're talking about.

I think I understood what you meant by reducing the top prime in the Legendre symbol, (p/q). One might as well start with p > q and then find what p is congruent to modulo q. That won't likely be a prime any more, but one can try factoring it to simplify the calculation even more.

After looking at Vanden Eynden's text, I found this relationship which might be interesting and is part of his (Gauss's) proof of QR.

Let p and q be odd primes with p > q and p is congruent to q mod 4. Then there exists some integer a such that p = q + 4a. Now that last equation implies the existence of three other congruence relationships.

1) If p = q + 4a, then p ≡ q (mod 4a). This is just the original congruence including a.
2) If p = q + 4a, then p = 4a + q and so p ≡ 4a (mod q). Now, 4a is linked to p via q.
3) If p = q + 4a, then -q = 4a + p(-1) and so -q ≡ 4a (mod p). Now 4a is linked to -q mod p.

In this way 4a is the link between p and q. One can get the QR result for the cases when p ≡ q (mod 4) by considering the following:

(p/q) = (4a/q) = (4a/p) = (-q/p) = (-1/p)(q/p)

The first part comes from p ≡ 4a (mod q). The second part was proved in the book and is non trivial, but can be assumed for the moment. The third part comes from 4a ≡ -q (mod p) and the last part comes from the multiplicative property of the Legendre symbol.

I'm not sure if this helps any, but it seemed interesting to me.

It is interesting. It is close to what I was doing last night on my own. Another interesting fact that could easily be overlooked is that when you square numbers mod (m) and every entry is duplicated once, it is an odd number and an even number which produce the same result in every case. What that means is factorization is not unique in this language. 52 and 62 both equal 3 (mod 11).

I was over at a math site recently and asked what they know over there. No responses yet, but I am sure they will not do as well as we are doing here. They love to talk a big game over there about advanced calculus and other impressive topics, but ol' QR is quite enough to stop all their chatter, espesially when I told them I was not interested in restatements of the law or facts surrounding it. Those surrounding facts I am interested in, but they don't need to know that, since we can get that right here and discover those facts for ourselves. I don't need them blabbimg restatements forever because they don't know what else to do. Maybe someone over there will come through yet. Don't hold your breath.

When it comes to predicting whether an overlapping square will be even or odd, forget about it, at least so far with what we know.

[desiresjab]

Oh, and by reducing I mean simply to carry out the division mentally and find the remainder to see if it is a square. 31 reduces to 1 (mod 3), for instance. Specifically we could say, 10p+1=q, or (q-1)/10=p. This is in the neighborhood of what you were saying above.

A relevant concept I came up last year is "highly even" and "barely even" numbers. 4n+1 numbers minus one are all highly even and 4n+3 numbers minus one are all barely even. A barely even number is divisible by 2 only once, a highly even more than once. This falls right out of something called the ruler function, which I also discovered in my investigations.

These two types of numbers are obviously germane to QR. The higher degree of evenness of Eisenstein's rectangle when both triangles have an even number of lattice points must certainly be an important fact. When QR =-1, the overall rectangle has a downright paucity of factors of two, managing onlyfour of them.

The real problem with the above idea is that two (4n+1)-1 numbers which by definition have a higher degree of evenness can reject each other, so to speak, and mutually not be in each other's quadratic residue set.

I really enjoy our discussion here. I am quite lucky to find even one able volunteer willing to go along on this perilous mission with me. The biggest problem I have is that I am wearing myself out thinking about it.

The layout of this forum is actually superior to the math forum I visited when it comes to typing math. This forum at least allows exponents. The other forum still uses dumb up arrows for powers.

[desiresjab]

Damned duplicate posts!

[desiresjab]

the 7x11 rectangle with 60 interior lattice points is only divisible by 2 twice. Another conjecture gone awry, perhaps. So far, no matter what I try to connect the behavior to, it only looks promising for a while.

But wait. Are all such rectangles whose dimensions are (4j+3)(4k+3) divisible by two only twice? Is that the nature of them? Yes, of course. What else would I be talking about? (Beat my own forehead). The train is still on track. One wheel anyway.

This paucity of 2's may lead directly to the mother lode, the reciprocity mechanism.

[Dreamwoven]

Damned duplicate posts!
Well, its better than the problem you had a while back, of disappearing posts!

[YesNo]

It is interesting. It is close to what I was doing last night on my own. Another interesting fact that could easily be overlooked is that when you square numbers mod (m) and every entry is duplicated once, it is an odd number and an even number which produce the same result in every case. What that means is factorization is not unique in this language. 52 and 62 both equal 3 (mod 11).

It looks like 3 is a quadratic residue because both 5 and 6 can be squared to give 3 mod 11. But then we know half of the elements will be quadratic residues and the other half non-residues. So there should be two elements when they are squared that give 3.

Another way of looking at this is to ask what solutions are there to the following polynomial: x2 - 3 = 0 (mod 11) There should be 2 solutions and there are.

When we are looking at the residue classes mod 11, we aren't looking at integers any more. Instead we are working with equivalence classes of integers, or sets of integers. The integers would be in the integral domain Z, where there are primes because multiplicative inverses for all elements do not exist, but these residue classes are in the finite field Z/Z11 where there aren't any primes anymore. In the finite field, the non-zero elements all have multiplicative inverses and so they would be units like 1 and -1 are in the integers Z. For example let n, not equal to 0, be an element in Z/Z11, then since n11-1 = 1 (mod 11), the multiplicative inverse of n is n9 (mod 11).




I was over at a math site recently and asked what they know over there. No responses yet, but I am sure they will not do as well as we are doing here. They love to talk a big game over there about advanced calculus and other impressive topics, but ol' QR is quite enough to stop all their chatter, espesially when I told them I was not interested in restatements of the law or facts surrounding it. Those surrounding facts I am interested in, but they don't need to know that, since we can get that right here and discover those facts for ourselves. I don't need them blabbimg restatements forever because they don't know what else to do. Maybe someone over there will come through yet. Don't hold your breath.

I got an account on https://math.stackexchange.com/ to get more information as well. It is good to have questions. The available answers aren't all the answers. Although QR is useful, what we really want is a quick way to evaluate (p/q) without having to consider (q/p). Quadratic reciprocity allows us to evaluate the one that is easiest to calculate, but perhaps there is a faster method. That sounds to me like what you are looking for.



When it comes to predicting whether an overlapping square will be even or odd, forget about it, at least so far with what we know.

That is an interesting question. One doesn't have to take the representatives for the equivalence classes from {0,1,...,p-1}. They could come from {-(p-1)/2,...,(p-1)/2}. The evenness and oddness of the result might change when using that set.

[desiresjab]

It looks like 3 is a quadratic residue because both 5 and 6 can be squared to give 3 mod 11. But then we know half of the elements will be quadratic residues and the other half non-residues. So there should be two elements when they are squared that give 3.

Another way of looking at this is to ask what solutions are there to the following polynomial: x2 - 3 = 0 (mod 11) There should be 2 solutions and there are.

When we are looking at the residue classes mod 11, we aren't looking at integers any more. Instead we are working with equivalence classes of integers, or sets of integers. The integers would be in the integral domain Z, where there are primes because multiplicative inverses for all elements do not exist, but these residue classes are in the finite field Z/Z11 where there aren't any primes anymore. In the finite field, the non-zero elements all have multiplicative inverses and so they would be units like 1 and -1 are in the integers Z. For example let n, not equal to 0, be an element in Z/Z11, then since n11-1 = 1 (mod 11), the multiplicative inverse of n is n9 (mod 11).




I got an account on https://math.stackexchange.com/ to get more information as well. It is good to have questions. The available answers aren't all the answers. Although QR is useful, what we really want is a quick way to evaluate (p/q) without having to consider (q/p). Quadratic reciprocity allows us to evaluate the one that is easiest to calculate, but perhaps there is a faster method. That sounds to me like what you are looking for.



That is an interesting question. One doesn't have to take the representatives for the equivalence classes from {0,1,...,p-1}. They could come from {-(p-1)/2,...,(p-1)/2}. The evenness and oddness of the result might change when using that set.

Huh? I am not looking for a faster way to do anything. I have said so many times what I am looking for that I do not feel like saying it again right now.

Something I said before is coming true. Hardly anyone understands quadratic reciprocity. No one is answering on the math forum I visited. They love to showoff. If anyone knew, they would surely answer. I feel partially vindicated in that the problem truly is difficult. Learning enough about it to pass a course in elementary number theory is not that hard, but knowing what I am asking--now that is hard.

To address something you said: In fact, the parity would change when we go to negative representatives of the residue system. Good observation.

[YesNo]

What question did you ask them?

[desiresjab]

What question did you ask them?

I said I was only intersted in the mechanism itself that made 4n+3 primes behave as they do toward each other in QR, not restatements of the law or facts surrounding it, and that I insisted on seeing this in terms of the numbers themselves rather than a higher abstaction coming out of group symmetries and the like. I admitted this might be like standing outside a forest with a flashlight looking for something hidden behind a tree, but still I asked for an explanation in terms of the numbers.

I suspect that the basis of some higher abstraction proofs in abstract algebra is to show an anti-symmetry between the two subrings of squares, or something along those lines. Those proofs look down from above, I want to look from below in the guts of the machine.

In the meantime I have developed what may be a valid conjecture, that the two triangles WAY and YAX in Eisenstein's rectangle will contain an identical number of lattice points when p and q are twin primes, and only then. This may seem obviious but be very hard to prove. It is an ad hoc conjecture, just a problem, like so many that Erdos proposed, perhaps not important, but a fact nonetheless and an interesting challenge to recreate upon.

A weakness of the conjecture is that I suspect that the outside members of prime triplets which are large enough might also do this. Okay, I leave out the if and only if part of the conjecture.

[desiresjab]

After merely checking all of my experiments on graphing paper, I am positive the conjecture could be extended to all prime triplets from at least the second such triplet to infinity with a stipulation. That stipulation is that two of the prime triplets are 4n+1 numbers, because when two 4n+3 primes clash we know they will obviously have different numbers of lattice points in the two triangles because one will be even and the other odd. As long as our prime triplets contain two 4n+1 numbers, we will be okay, and I am pretty sure that conjecture would hold.

[YesNo]

In the meantime I have developed what may be a valid conjecture, that the two triangles WAY and YAX in Eisenstein's rectangle will contain an identical number of lattice points when p and q are twin primes, and only then. This may seem obviious but be very hard to prove. It is an ad hoc conjecture, just a problem, like so many that Erdos proposed, perhaps not important, but a fact nonetheless and an interesting challenge to recreate upon.

A weakness of the conjecture is that I suspect that the outside members of prime triplets which are large enough might also do this. Okay, I leave out the if and only if part of the conjecture.

That sounds like an interesting problem. What do you mean by prime triplets? Three consecutive primes?

[desiresjab]

That sounds like an interesting problem. What do you mean by prime triplets? Three consecutive primes?

There are two different brands for p, q and v, {n, n+2, n+6} and {n, n+4, n+6}.

The conjecture is that as long as at least two in either set are of the form 4n+1, they will obviously express not only the same parity with each other in all three combinatory pairs, but also have the same number of lattice points in WAY and YAX, as seen from Eisenstein's rectangle represented in Wiki-pejia.

It seems intuitively clear, but I don't know how to prove it. I am not saying that is the maximum boundary condition, either. That is, there may be wider gaps than six which allow for an identical number of lattice points in both triangles. It depends on only two things--the absolute cardinality of p,q and v, and whether only one of them is a 4n+3 number.

The larger the absolute magitude of the triplet, the closer any two of them compared to each other will be to the ratio of a square 1/1. Since this holds for even small triplets where the ratio is not as close to 1, it must be true for ones of larger absolute magnitude that meet the only other condition.

What I suspect could be proven with analytical methods is that for any gap, as wide as one wants to make it, WAY and YAX can still produce the same number of lattice points, as long as the primes involved are large enough and both are not type 4n+3. Similar things have been proven about primes and other intervals. This one feels intuitively right. It might well have been already proven, or at least conjectured.

[YesNo]

After sleeping on your problem related to twin primes, I realized I couldn't solve it.

It would be easy to show that the parity in the two triangles of lattice points are the same, but not that there are exactly as many lattice points in both triangles. In the triplets, having two 4n + 3 primes would make the number of lattice points in the rectangle odd and so the parity in the triangles would be different.

One way of solving it might be to go through Eisenstein's construction using a prime p = 4m + 1 and then seeing if the lattice points remain the same using 4m + 3. I realize I don't understand Eisenstein's proof well enough to do this easily.

Also one might be able to generalize this to any two numbers whose difference is 2. Some of the points might be on the diagonal line separating the two triangles, but then they would either not be counted or counted in both triangles.

If you haven't published this problem, it might be worth doing so say on places like math.stackexchange.com. An interesting question is more valuable than a quick solution.

[YesNo]

I was thinking more about the number of lattice points in the two triangles. Intuitively, it would seem that there should be the same number of lattice points in both triangles because the diagonal line divides the rectangle into two equal area triangles. With two 4m + 3 primes the total number of lattice points in the rectangle is odd, so one of the triangles should have an odd number of lattice points and the other an even number. There has to be a difference of at least 1 for those primes.

Do you have an example (two primes p and q) where the difference in the number of lattice points is greater than 1?

Maybe the more basic question is to find pairs of primes that divide the lattice points in the two triangles so that the difference in the number of lattice points in each triangle gets larger.

[desiresjab]

I was thinking more about the number of lattice points in the two triangles. Intuitively, it would seem that there should be the same number of lattice points in both triangles because the diagonal line divides the rectangle into two equal area triangles. With two 4m + 3 primes the total number of lattice points in the rectangle is odd, so one of the triangles should have an odd number of lattice points and the other an even number. There has to be a difference of at least 1 for those primes.

Do you have an example (two primes p and q) where the difference in the number of lattice points is greater than 1?

Maybe the more basic question is to find pairs of primes that divide the lattice points in the two triangles so that the difference in the number of lattice points in each triangle gets larger.

Examples are easy to come by for 5 and 11, the triangles have 6 and 4 lattice points. The difference always has to be at least 2 for examples of the same parity.

For primes 5 and 15, they are 9 and 7.

It has not failed for twin primes or correctly composed prime triplets, and I have not found it to be true anywhere else.

* * * * *

What I have done with regards to QR is moved on to abstract algebra. Fortunately for myself, I can relate a great deal of what they are saying to what I already know of groups, rings and fields from number theory, otherwise I would be lost. In lecture 27 of 38 they finally got real close to QR.

[YesNo]

Examples are easy to come by for 5 and 11, the triangles have 6 and 4 lattice points. The difference always has to be at least 2 for examples of the same parity.

That's good to know.



For primes 5 and 15, they are 9 and 7.

Although 15 is not prime, I would think this should work for numbers in general. Some of the lattice points could be on the diagonal line if one doesn't use primes.



It has not failed for twin primes or correctly composed prime triplets, and I have not found it to be true anywhere else.

I wonder if it is possible to pair the columns of lattice points. For example, the column where x = 1 would pair with the column where x = (p-1)/2. I suspect the number of lattice points from just these two columns would be the same in both triangles. Then proceed by induction, or some other means, to look at the other column pairs.



What I have done with regards to QR is moved on to abstract algebra. Fortunately for myself, I can relate a great deal of what they are saying to what I already know of groups, rings and fields from number theory, otherwise I would be lost. In lecture 27 of 38 they finally got real close to QR.

Which text are you using? Youtube may also have interesting reviews of algebra.

[desiresjab]

That's good to know.



Although 15 is not prime, I would think this should work for numbers in general. Some of the lattice points could be on the diagonal line if one doesn't use primes.



I wonder if it is possible to pair the columns of lattice points. For example, the column where x = 1 would pair with the column where x = (p-1)/2. I suspect the number of lattice points from just these two columns would be the same in both triangles. Then proceed by induction, or some other means, to look at the other column pairs.



Which text are you using? Youtube may also have interesting reviews of algebra.

I meant 5 and 17. I believe there was one pair of primes that gave 9 and 6, but I cannot remember what the[pair was.

Presently, I do not have an abstract algebra text. I am doing everything off the internet.
In case you want a link to the thirty-eight lectures, I will provide it below. The instructor's name is Gross from Harvard, and this guy is exceptional.

Abstract algebra is a whole new language. They operate at a very high level of abstraction. If you miss somethibng, you have to go back. Everything is dependent on what is already supposed to have been learned. Very abstract thinks like manipulating and untangling compositions of functions. You have to know a kernel from an image, you constantly have to check to make sure what you are working on is actually associative or commutative, et al.

He may get to QR the next lecture I have up. What I can tell you is the solution wull be buried under even more layers of abstraction than I imagined.

I try to go too fast--six or seven lectures per day or more. That guarantees I will have to go back and do it agin. But by going ahead to the end, I know exactly what I should be concentrating on the second time around. The method is not as faulty as it seems.

https://www.youtube.com/watch?v=TsLWvWf4RdY&spfreload=10#t=6.688125

[YesNo]

I think I found it: https://www.youtube.com/watch?v=EPQgeAz264g&list=PLA58AC5CABC1321A3

[desiresjab]

I think I found it: https://www.youtube.com/watch?v=EPQgeAz264g&list=PLA58AC5CABC1321A3

That is the right guy. Benedict Gross. I think he is chairman of the department of mathematics at Harvard. He never does get to QR, though there is a fair amount of material about squares, since that was a big subject of Gauss. The best minds today are still working on the stuff developed out of Gauss, who set the course for everything in the field and has never had a conjecture overturned anywhere in math or science but many verified.

Instead of numbers, these people study equivalence classes of numbers and of polynomials, both real and complex, through structures like groups, rings and fields. Every structure is carefully defined, and one has to know them apart and be aware of the criteria for being in one or the other during any process.

* * * * *

If a ground level view of the mechanism for 4n+3 primes in QR is possible, maybe someday I will find it. The tools and results of abstract algebra are often macroscopic but probably capable of focusing down to the particular mechanism responsible, too.

[desiresjab]

No sooner do I give up than I see what seems to be enough. It is like I said before, if p is a 4n+3 prime, p-1 will be divisible by 2 only once. The mechanism is best described for illustration in my own terminology as the difference between "barely even" and "highly even" even numbers. On one of those classes is where you end up once you subtract 1 from p.

I had to look at it long and painfully to finally confirm that what was staring me in the face was the actual mechanism itself. That is why I am dumb.

All that remains is to verify and confirm that one understands how we got to the point of (p-1)/2 in the first place. That is incredibly easy, already done. I am finished.

[desiresjab]

Let me sum up, try to anticipate any questions, and move these results back into a discussion of general cosmology.

The whole giant digression involving QR took place because I wanted to take the proposition that God could not make a universe where 2 is not the successor of 1 to a higher level, on a road to what might even include all of mathematics, but at minimum enlarging the statement to God could make no universe where the statements of mathematics would be false, from at least the founding axiom through to somewhere beyond the law of quadratic reciprocity. Anything theoretically true here would be theoretically true in any other universe as well, and vice versa.

No such universe is imaginable, might be a less religiously provocative way of stating it. Now that I can state it, there is only to wrap up the discussion of QR and return to cosmology, where the above will be one of the postulates of my personal philsophy within cosmology--mathematical cosmology, I suppose.

A gem that comes out of this is that we have the capability to understand any universe, and any universe of any description, no matter how different from our own, would have the capability of understanding our universe. That capability, consisting of mathematics and its growing extensions, would remain invariant across universes, while retaining its elastic variability and variety.

* * * * *

Eisenstein's P by Q rectangle must be viewed as a scaling object, just as if we had two gears intermeshing with radii corresponding to the lengths of the two 4n+3 primes, it expresses their ratio. I think his rectangle must be the simplest scaling object for this problem.

As I suspected a while back, the problem is a paucity of 2's when both primes are 4n+3. Once the rectangle is divided into four quadrants, there is no 2 left over, in other words, each quadrant contains an odd number of lattice points. That paucity of 2's forces the diagonal to cut the smaller rectangle of WAYX into two unequal halves of different parity.

The dimensions of the interior rectangle within ABCD containing only interior lattice points is

(p-1)(q-1).

This is none other than Euler's phi function Φ(p), the measure of how many numbers less than p are prime to it. There are sixty lattice points in the interior of ABCD. Points on the perimeter are not prime to one or other prime, so cannot be included. As usual, (mod 0) not allowed.

Once odd primes have the extra freedom of at least one more factor of 2, the problem is resolved, and the two primes are forced to act together, forced to the same parity because their smaller rectangle contains an even number of lattice points.

At the beginning, Eisenstein calculates the number of even lattice points in ABCD. He knows if they are aysmmetrical in the two halves of ABCD, so will the odd lattice points be, to make up for the discrepancy. The number of even lattice points in ABCD is of course thirty. His initial additive method was good enough for the parity of p, (-117), but not good enough to obtain q's parity.

(P-1/2)(q-1)/2) is indeed the total number of points in WAYX, 1/4 of the total points in ABCD, and 1/2 the number of even points.

-1(5)(3) is -1(60/4) or -1Φ(p)/4 after all. So Eisenstein's exponents are correct not only in their parity but they are also the "appropriate" exponents in that they are a factorization of the number of lattice points in WAYX. More importantly, (p-1/2) and (q-1/2) are the number of quadratic residues for each prime, we already know.

Now, that is everything about Eisentein's rectangle. Does it really prove quadratic reciprocity?

Yes, here is why I think so. Since (p-1/2) and (q-1/2) are a simple count of the number of quadratic residues for each prime, and their product is used as an exponent to count -1 back and forth from negative to positive, both are represented, and their product forms the dimensions for the rectangle of inner points of WAYX, and the exponentiation's result can only be negative when both (p-1/2) and (q-1/2) are odd.

For me that quite settles the issue, not only for 4n+3 prime pairs but for any prime pair, excluding 2, which I can recite the forumla for but have not yet reasoned out. My focus has been the 4n+3 primes, knowledge of which I hoped would illuminate the triggering mechanism for other prime pairs as well, which has happened for me.

4n+3 primes have to be the same thing in any universe. The only fundamental difference between 4n+1 and 4n+3 numbers is the degree of evenness when you subtract 1 from them.

* * * * *

Now that we all know this fundamental fact of numbers and how it constrains universes, we can proceed with broader cosmology again.

[YesNo]

A gem that comes out of this is that we have the capability to understand any universe, and any universe of any description, no matter how different from our own, would have the capability of understanding our universe. That capability, consisting of mathematics and its growing extensions, would remain invariant across universes, while retaining its elastic variability and variety.


I agree that given the initial axioms, the mathematical results are invariant across all possible universes. That there exist other universes can be assumed based on knowing that our universe is not eternal. In particular, the big bang shows it had a beginning.

Edit: Regarding the "barely even" number, those having only one factor of 2, there is a concept called "singly even" or "oddly even" that matches that: https://en.wikipedia.org/wiki/Singly_and_doubly_even

[desiresjab]

I agree that given the initial axioms, the mathematical results are invariant across all possible universes. That there exist other universes can be assumed based on knowing that our universe is not eternal. In particular, the big bang shows it had a beginning.

Edit: Regarding the "barely even" number, those having only one factor of 2, there is a concept called "singly even" or "oddly even" that matches that: https://en.wikipedia.org/wiki/Singly_and_doubly_even

I like highly even and barely even better. I modeled it after Ramanujan's idea of highly composite numbers, which assigns an index to each number to rank how composite it is. With the ruler function I can calculate and assign any even number its degree of evenness, without calculating those that precede it.

A few last details to focus up.

For the 7x11 rectangle, Eisenstein's exponents of 5 and 3 give a correct factorization of the lattice points in WAXY, but for the 7x3 rectangle where there are three points in WAXY, the exponent is 3, which does not factor the lattice points of WAXY, but sums them. Both the additive and the multiplicative coincidences may have been just that. I am not worried about that, it is minor and will sort itself out.

I saw the mechanism behind the behavior of 4n+3 primes from ground level. Mission accomplished. But not quite. For of slightly more consternation is that I understand the proof but not why it proves QR. True, WAXY is a quadrant, but the last bit of "seeing" has not clicked into place as to how this proves whether or not p or q are in each other's quadratic residue sets.

Can I not simply say Φ/4, where Φ is Euler's totient function, divided by four will always give the correct number of points in WAXY, and be done with it? I believe I can. This has to work for both species.

I understand everything about this proof except why it proves what it proves.

[desiresjab]

In the meantime, I see clearly that Φ/4, where Φ is Euler's totient function, will always give the correct number of lattice points in WAXY. This may work for 4n as well as 4n+2 numbers. I find this connection with the totient function highly intriguing.

[Φ(pq)]/4, in other symbols. Φ(pq) is what the RSA encryption system is based on. It is extremely hard to find Φ(pq) unless you know what p and q are. But you only know what pxq is, which means you have to find its factors, and that is almost impossible for huge, "barely composite" numbers with today's computing technology and math techniques.

[desiresjab]

Here is a very interesting near hit for the rectangle 11x7.

Φ(77)=60, Φ(60)=16, Φ(16)=8.

If Φ(60) had been 15 instead of 16...? Well, that is how research begins, I guess. But what if there is a pattern in the descending chain of Φ's anyway that only more investigation will ferret out? It should be easy to look at other examples, but my brain is shutting down right now. More later.

[desiresjab]

Why, looky here.

Φ(7x3)=12→ Φ(12)=4→ Φ(4)=2.

The reduction by a chain of Φ's instead of by factors of 2, worked out exactly as before, probably close enough to warrant even further investigation.

The second Φ again equaled one more than the total number of lattices in WAXY; the third Φ equaled the greatest number of lattices in either of the triangles. Uh-oh, now we have to go on.

Φ(19x23)=396

Φ(396)=120

Φ(120)=32.

Here everything goes quite amiss and we see we are on a deadend using the chain of Φ's. It was merely another coincidence. But Φ/4 is not a coincidence. That baby is real and will get you the right parity and the correct number of points in WAXY, which in this case is 99.

Φ/4 rules, Φ of Φ of Φ does not. Now we know for sure. I do not think 120 is one more than 99, and I do not think 32=49. That case closed.

[desiresjab]

It seems wonderful, it is wonderful. Why do Φ and these squares interact at all?

What do Φ(pq)=(p-1)(q-1)=a, or for that matter pq=z itself, and rt=d, the distance formula, have to do with each other? They are in some kind of equivalence class with V=IR, the voltage formula, and the equations for uncountable (literally) other phenomena. In abstract algebra they are called isomorphisms. That means they are just re-lablings of each other. They have the same group characteristics. A popular way of saying it is: they are the same under the hood. Their matrices look identical except for the difference in labels. Isomorphism. Of all those creepy isms in abstract algebra they are the easiest to see clearly. Just when you think you see automorphism or homomorphism clearly, they add some more bugaboo to it or manage to make them unclear by other means.

(p-1)(q-1) just happens to belong to this equivalence class of functions, linear relationships known as directly proportional to. Something like that. One cannot help noticing the similarity of the equations for work done by two workers doing a job and resistance from two resistors in parallel in electronics, as another example. The phenomena of the world around us express certain classes of functions replicated endlessly with only different labels under the hood. A few basic classes dominate much of the action, it seems to me at this moment.

I guess I will continue to loiter around the QR lobby until I see why Eisenstein has proved it. I see everything else about his beautiful proof, it has illuminated the operative principle of both 4n+3 primes and 4n+1 primes in QR by rectangular illustration, it provides exact numbers, I might as well hang around to see the reason it does what it purports to do.

I said I would be happy if I could see the general mechanism, and I have, but I guess I lied, for I am not satisfied now until I can see what makes Eisenstein's rectangle a proof of QR. All of it is right on the page and obvious the way math always is, a grand tautology, so eventually it will pop out clearly, the way the mechanism did, after my staring at it ignorantly forever. At the next moment of revelation I am bound to see more clearly something I have already stated, if past is precedent.

[YesNo]

But Φ/4 is not a coincidence. That baby is real and will get you the right parity and the correct number of points in WAXY, which in this case is 99.

If p and q are distinct primes then Φ(pq) = Φ(p)Φ(q) = (p-1)(q-1). All we need is to divide by 4 to get the number of lattice points. So I agree Φ/4 will always equal the number of lattice points in Eisenstein's rectangle.

[YesNo]

said I would be happy if I could see the general mechanism, and I have, but I guess I lied, for I am not satisfied now until I can see what makes Eisenstein's rectangle a proof of QR. All of it is right on the page and obvious the way math always is, a grand tautology, so eventually it will pop out clearly, the way the mechanism did, after my staring at it ignorantly forever. At the next moment of revelation I am bound to see more clearly something I have already stated, if past is precedent.

It is good you are not satisfied. Otherwise you would stop looking.

One of the problems with both Gauss's and Eisenstein's proofs is that they do not directly show why (p-1)(q-1)/4 should be the exponent. The direct proofs are in the Gauss Lemma and the Eisenstein Lemma. Those exponents in those lemmas are different from (p-1)(q-1)/4. But they want the exponent to be (p-1)(q-1)/4 because that is computationally easier to work with and so they transform their results so that the parity is preserved, which is all I think they are interested in.

At least that is how I see it at the moment.

[desiresjab]

It is good you are not satisfied. Otherwise you would stop looking.

One of the problems with both Gauss's and Eisenstein's proofs is that they do not directly show why (p-1)(q-1)/4 should be the exponent. The direct proofs are in the Gauss Lemma and the Eisenstein Lemma. Those exponents in those lemmas are different from (p-1)(q-1)/4. But they want the exponent to be (p-1)(q-1)/4 because that is computationally easier to work with and so they transform their results so that the parity is preserved, which is all I think they are interested in.

At least that is how I see it at the moment.

I feel that 3 & 5 must be the natural exponents. The only way to get lower is to have 1 & 3, like the 7x3 rectangle. Also there is the connection of Φ to them.

I do not believe the sum of the two exponents has much bearing. The exponents using Eisenstein's algorithm are 1 & 3 for the 3x7 rectangle. Those exponents do the job, Φ/4 does part of the job even faster, but it does not distinguish between p and q, as the exponents do.

Something just became slightly more focused in my head that is still blurred. If (p-1) & (q-1) were the same number and we multiplied them together we would be squaring, in which case we would merely add their exponents, wouldn't we? This is why adding exponents cannot work here. But the symmetries between squaring and what we are doing is intriguing. One feels that box might contain some secrets.

Several things we know for sure: Two sure ways to the correct result are Eisenstein's algorithm and Φ/4. It just happens that the intermediary step Φ/2 is the number of quadratic residues of each prime, and the final step the dimensions of the rectangle inside WAXY the lattice points sit on.

I am beginning to see that QR is as centrally connected as its reputation says it is. These connections are only the tip of the iceberg, QR reaches into everything. No doubt there are proofs that exploit the connection to Φ, just as there are trigonometric proofs, proofs that exploit the Pythagorean theorem, proofs that exploit Fermat's little theorem, and all kinds of proofs from group theory and abstract algebra, including at least one vector proof.

It will fall into place, but not without more staring and effort. I am hoping for a few weeks or less.

Something about 5x3 is really nagging me.

[YesNo]

Something just became slightly more focused in my head that is still blurred. If (p-1) & (q-1) were the same number and we multiplied them together we would be squaring, in which case we would merely add their exponents, wouldn't we? This is why adding exponents cannot work here. But the symmetries between squaring and what we are doing is intriguing. One feels that box might contain some secrets.

From one direction, we really should be adding the exponents to get the result. That is where Gauss's and Eisenstein's lemmas start. They compute an exponent for each (p/q) and (q/p) with -1 as the base. Say these are u and v respectively. Then (p/q)(q/p) = (-1)u + v.

From the other direction, the product gives the desired result. That is the result Legendre observed when he conjectured the result depended on whether p and q were congruent to 1 or 3 modulo 4. Note that (p-1)(q-1)/4 is just the statement of the desired parity using p and q mod 4.

To connect the two directions are transformations from the sum of those exponents to that product that preserves parity.

[desiresjab]

From one direction, we really should be adding the exponents to get the result. That is where Gauss's and Eisenstein's lemmas start. They compute an exponent for each (p/q) and (q/p) with -1 as the base. Say these are u and v respectively. Then (p/q)(q/p) = (-1)u + v.

From the other direction, the product gives the desired result. That is the result Legendre observed when he conjectured the result depended on whether p and q were congruent to 1 or 3 modulo 4. Note that (p-1)(q-1)/4 is just the statement of the desired parity using p and q mod 4.

To connect the two directions are transformations from the sum of those exponents to that product that preserves parity.

I think we could be on the right track with these thoughts. Hell, maybe you have already seen it through and through. All I know is I haven't.

[YesNo]

I don't know much about quadratic reciprocity. I haven't figured it out. Besides, once we get past quadratic reciprocity, there's cubic, and then quartic and then on and on for all I know. So we have only skimmed the subject.

I have been trying to put together the Artin's Conjecture puzzle. I'm starting with 2 and trying to find ways to get an infinite number of primes for which 2 is a primitive root. I think it is easy to get an infinite number of composite numbers for which 2 is a primitive root: just take an odd prime p for which 2 is a primitve root, then it should be a primitive root for pn if (2/p) = -1 (mod p2). Actually, I am not sure that is right, but it doesn't solve Artin's Conjecture. That requires an infinite number of primes, not composites.

To keep my motivation, I have started asking questions on math.stackexchange centered around trying to show that there are an infinite number of Germain primes. These are primes p such that 2p + 1 is also a prime. They are easier to work with. If there are infinitely many of them that should solve Artin's Conjecture for a = 2.

[desiresjab]

I don't know much about quadratic reciprocity. I haven't figured it out. Besides, once we get past quadratic reciprocity, there's cubic, and then quartic and then on and on for all I know. So we have only skimmed the subject.

I have been trying to put together the Artin's Conjecture puzzle. I'm starting with 2 and trying to find ways to get an infinite number of primes for which 2 is a primitive root. I think it is easy to get an infinite number of composite numbers for which 2 is a primitive root: just take an odd prime p for which 2 is a primitve root, then it should be a primitive root for pn if (2/p) = -1 (mod p2). Actually, I am not sure that is right, but it doesn't solve Artin's Conjecture. That requires an infinite number of primes, not composites.

To keep my motivation, I have started asking questions on math.stackexchange centered around trying to show that there are an infinite number of Germain primes. These are primes p such that 2p + 1 is also a prime. They are easier to work with. If there are infinitely many of them that should solve Artin's Conjecture for a = 2.

I have put my ten thousand hours of music in; I have put my ten thousand hours of writing prose and poetry in. With these enterprises I did much more than that. But I must be at about five thousand hours of math because the moves are not instinctive the way they would be with professional mathematicians. It takes me a long time, I make a lot of mistakes and move in with false assumptions quite often that later embarrass me.

For we amateurs maybe that is the way of it. Most unsloved problems scare me off because I know my inabilty to resist temptations. I only move in with a problem when it advances the cause of advancing my learning, or of it has a particular form that intrigues me. Sometimes they have been fun problems that I found here or there; I got into criptorithms for a while. Most of the time now I move in where I think I can learn as much as possible.


I have lived with Fermat's little theorem, Euler's phi function and quadratic reciprocity in my time. I am going to court Euler's divisor functions coming soon, because the few kisses I stole were not enough. That girl has a lot to say. Very centrally connected. After that I have to move in with complex numbers for a long time. Meta mathematics has been going on there since Gauss formalized the language.

I need two minds or more. The other side has now pulled me back to my fictional trilogy. That enterprise is where quadratic reciprocity is--awaiting the final clarity of events that make everything fit. The series is done except for some middle chapters left out in the inspirational heat of moving forward.

[YesNo]

On the one hand I am not trying to solve Artin's Conjecture. I'd be happy with the subjective process of understanding it before moving onto something else. On the other hand I feel like a teenager looking at the stack of books in the library and planning on how to read all of them. What probably counts is the subjectivity involved in understanding something.

Have you ever posted your question on a mathematics forum about the number of lattice points in Eisenstein's triangles? That sounds like an interesting puzzle. I haven't heard it mentioned except here, but then I don't know much about lattice points. People only started counting lattice points a few decades ago based on some cursory research I did into your problem.

Regarding those lattice points, if p = q then the two triangles should have the same number of lattice points since the slope of the diagonal would be 1 and it should go through all the lattice points on the diagonal. Letting q = p + 2 keeps that slope close to 1 and the diagonal misses all the lattice points. That's how I'm looking at the problem at the moment.

[desiresjab]

Within a short while I will see the final piece. I don't need them now.

[YesNo]

Does anyone participate in grid computing volunteer projects by letting those projects use their computer's resources in the background?

I understand some of them allow one to participate in searches for extraterrestrial life or pulsars or even special prime numbers. Here is the site: http://boinc.berkeley.edu/download.php

[Dreamwoven]

I think that is to give access to your computer for them to use it while you sleep or are away.

[YesNo]

Yes, they would use it for some project you signed up for. It could be something related to cosmology (like gravitational waves) as well as astronomy or health or even mathematics (finding primes).

I haven't signed up for it, but I do have an old computer that I might as well clean up and turn on for them to use.

It is sort of like a large communal gaming system that might have some use-value besides the game itself.

[YesNo]

I started up the old laptop, downloaded the BOINC software, installed it as administrator, signed up for PrimeGrid and the computer is now happily busy again. Well, I don't know if it is happy about that, but I am glad to put it to use.

[YesNo]

I started reading Jimena Canales, "The Physicist and the Philosopher: Einstein, Bergson, and the debate that changed our understanding of time".

I am hoping it will help me understand the cultural context in which we view cosmology today. The debate mentioned in the book occurred on April 6, 1922, and apparently has cultural influence to the present day. I hadn't heard of it before, but we don't have to be aware of the influences we have, especially those we take for granted.

[desiresjab]

My conclusion is that Eisenstein already knew what he was after. He was after a way of arriving back at Euler's criterion through his lattice point rectangle representation, which he managed to do. He shows that the number of even lattice points in ABC has the same character as the total points in CZY which is no different from AYX. Then it is seen that WAY is also equal to the unmarked triangle in his diagram that sits under CZY.

He shows that his exponent u+v is equal to (p-1/2) (q-1/2). Very good. The deed is done.

However, it does not show or explain the mechanics of why moduli behave toward each other according to species, landing or not landing in one another's quadratic residue sets. You cannot get those mechanics from this proof. It is the wrong kind of proof for that, it does not delve there. Viewing those mechanics may require learning some new mathematics. I am far more interested in the mechanics than the proof itself, it turns out. I need to see different proofs to determine which one explains the mechanics I am after.

The same mechanism seen in Eisenstein's proof that clearly explains the behavior of 4n+3 primes and pinpoints the reason for it does not explain the quadratic behavior of moduli in general. Without foreknowledge I believe there is no way Eisenstein could have worked backwards from his diagram to explain that quadratic reciprocity was merely Euler's criterion.

[YesNo]

There is only so much one can get from any one proof. Then one has to look elsewhere for other interesting ideas.

I put together a Google sheet trying to test your conjecture that the number of lattice points in the two triangles for twin primes are equal. It looks like it works for twin primes less than 100. That is all the further I tested it.

Another topic that has caught my attention are the lengths of prime gaps. For twin primes the length would be 2, however, the gap could be arbitrarily large.

[desiresjab]

There is only so much one can get from any one proof. Then one has to look elsewhere for other interesting ideas.

I put together a Google sheet trying to test your conjecture that the number of lattice points in the two triangles for twin primes are equal. It looks like it works for twin primes less than 100. That is all the further I tested it.

Another topic that has caught my attention are the lengths of prime gaps. For twin primes the length would be 2, however, the gap could be arbitrarily large.

There is any gap as long as you please if you go far enough out the number line. That has been proven. That proof must be a hard nut.

[YesNo]

There is any gap as long as you please if you go far enough out the number line. That has been proven. That proof must be a hard nut.

Yes. The proof that the gap can be arbitrarily large is well known. That there are infinitely many twin primes has not been solved. That's the hard problem. Here is a status from a Wikipedia article: https://en.wikipedia.org/wiki/Twin_prime


On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.[1][2] Zhang's paper was accepted by Annals of Mathematics in early May 2013.[3] Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang’s bound.[4] As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, the bound has been reduced to 246.

So there are infinitely many gaps of size as small as 246. One just has to get that down to 2.

[YesNo]

I started up the old laptop, downloaded the BOINC software, installed it as administrator, signed up for PrimeGrid and the computer is now happily busy again. Well, I don't know if it is happy about that, but I am glad to put it to use.

The computer crashed a few times, but I finally realized I should run it at 50% capacity rather than 100% capacity to give it a chance to cool off. Also I put it on top of small objects so that the heat can move away faster. But I got my first badge for running 10,000 points worth of stuff.

[Danik 2016]

Just trying to help with the posts. I am new at this forum and I myself lost some texts because this forum has a time out problem. Now, if it is a small post like this one I keep saving and editing it on the edit pad. If it is a long post like yours I write it on Word and then paste it on the forum page.
I lost another long post because of the idiotic setup of this forum. I am about done with this goat hole. It does not matter if I login first or not, it always tells me I do not have permission to post when I try to send my post, and I have to go through some other crap. Sometimes I have lost the post in the process. The people who run this outfit need to explain themselves.

Anyway, that was a great link. Right now I do not feel like trying to recreate my detailed post, so I will let it go for now.

[desiresjab]

Just trying to help with the posts. I am new at this forum and I myself lost some texts because this forum has a time out problem. Now, if it is a small post like this one I keep saving and editing it on the edit pad. If it is a long post like yours I write it on Word and then paste it on the forum page.

My problems disappeared, perhaps yours will too.

[YesNo]

I started reading Jimena Canales, "The Physicist and the Philosopher: Einstein, Bergson, and the debate that changed our understanding of time".

I am hoping it will help me understand the cultural context in which we view cosmology today. The debate mentioned in the book occurred on April 6, 1922, and apparently has cultural influence to the present day. I hadn't heard of it before, but we don't have to be aware of the influences we have, especially those we take for granted.

After reading a couple chapters in this, I realize that Einstein was involved in two conflicts. One of them was with Bohr over quantum physics and the other was with Bergson over the reality of time. In both, Einstein won the reputation prize since he is remembered better than Bohr and Bergson, but he lost the quantum physics debate to Bohr and, as I am beginning to see, he likely also lost the time debate to Bergson. However, with Bergson, I don't understand the issues at stake as well. They involve time dilation and the reality of various measurements, but that is as far as I've got. This question relates to a cosmology thread in that it questions the "reality" of "space-time".

[YesNo]

I am over half way through Canales' history of the debate between Bergson and Einstein.

I realize I would likely be on Bergson's side. By putting time and space together into "space-time" Einstein created a deterministic block universe where nothing new could happen. This doesn't fit reality as I experience it. So, being pragmatic, I assume there's something wrong with it.

Also it looks like Poincare and Lorentz, who came up with the measurable effects of relativity theory before Einstein did might have better ways of interpreting it, but I am still trying to figure out what those different interpretations are.

At a high level, the difference between Bergman and Einstein is obvious. Einstein gets a mathematical theory (from Lorentz) and then assumes that his preferred model is reality rather than just a way to model reality. Time is linked to space, because it is convenient for the mathematics to manipulate it that way. It is sort of like following Galileo and saying the Sun really is the center of the universe regardless of our current view that that is no more true than saying the Earth is the center of the universe. On the other hand Bergson is interested in presenting the lived experience of time which is not deterministic.

Canales did a good job of bringing in the various people who participated in this debate in the 20th century. I didn't realize how connected all of these ideas were.

[YesNo]

I finished the book a couple of days ago. I will probably have to read it again after it all settles.

The main problem in the book is whether time is really a succession of infinitesimal instants such as a point on a mathematical line or whether it is something with duration that we access through our subjectivity moving from a past to a future. For practical purposes, like synchronizing clocks, there is use-value in modeling time as a point on a mathematical line. That is not the question. The question is whether time really is a dimension of infinitesimal time-points linked to three dimensions of space-points called "spacetime". Einstein claims that spacetime is real. Bergson claims that the model has use value, but nonetheless it is a mathematical fiction falsified by our own experience of time.

One of the consequences of believing in spacetime is that the universe is then a "block" with four mathematical dimensions in which nothing happens. Both the future and the past are illusions. There is no "arrow of time". Nothing evolves. Why would that be the case? Because the mathematical equations used to model the universe don't change. Einstein promoted these equations from Lorentz' relativity model to reality itself. This allowed time to be reversible in Einstein's view of reality.

Belief in spacetime implies belief that light is the maximal speed and that it is a mathematical (and physical) constant. It is not just that it is convenient to view light as constant, but that light really has a constant speed, not only today but throughout the history of the universe. Indeed it is convenient to make that assumption. Around 1900 people were looking for something that didn't change on which they could base measurements of both length and duration. They couldn't find anything until some decades earlier it was discovered that light had a finite speed and then that it looked like we could not detect a difference in its speed.

Belief in spacetime is just one interpretation of relativity. Relativity itself predates Einstein's deterministic interpretation. Lorentz and Poincare had the mathematical model of special relativity prior to Einstein and neither Lorentz nor Poincare accepted Einstein's interpretation of it. What that means is that we can have relativity without being forced to accept Einstein's block universe determinism. What I wonder is to what extent this also applies to general relativity.

Finally there is the media problem with relativity. Do we (that is you and me as people hearing the scientific discussions) accept a scientific interpretation on rational grounds or because it has been promoted in the media with questionable rhetoric? Is science for us a political event? In the case of Einstein he went out of his way to promote his interpretation in the media and he used ad hominem arguments against those opposing his views suggesting that they were too stupid to understand him or that they were antisemitic. Now Bergson was also Jewish, so the debate wasn't about antisemitism. I doubt that the people who disagreed with Einstein were any more stupid than the people promoting Einstein's own interpretation.

As a conclusion, I think Einstein's deterministic block universe speculation has been falsified both by quantum physics and by the progression of living organisms from birth to death. Time, whatever it is, is not reversible. Hence Einstein's interpretation of Lorentz' original relativity theory is false. Nor is it needed to keep the benefits of relativity theory. Also, the politics involved in that discussion makes me wary of the politics involved in scientific discussions that we hear today.

[Dreamwoven]

This is an interesting philosophical argument. What do you mean by politics in your last sentence? Alternatively, what would not be a political argument that would be acceptable?

[YesNo]

One probably cannot avoid politics in discussions about science. On one level it involves which group of scientists get a bigger share of limited funding. On another it involves which view of reality will dominate our own common sense.

For example, I've been aware of the name Einstein since a child. I didn't even hear of Bohr until a few years ago when I started looking at quantum physics. I didn't know the name Bergson until I read this book. Although I heard the name Lorentz because of the mathematics grounding relativity, he appeared more as a footnote in the theory of relativity. It was his idea. Why is Einstein so front and center in my common sense? He didn't win his debate against Bohr. He shouldn't have won any debate against Bergson. That is probably not a "politics" in the strict sense of who will govern, but it affects who governs my common sense just as a political candidate might and the ad hominem rhetorical techniques to promote one person over the other seem similar.

[YesNo]

I am reading John Derbyshire's "Prime Obsession". He tried to explain the Riemanann Hypothesis in such simple terms that any one could understand it. I'll have to see if he was right. At one point when he was explaining the derivative he made this statement (page 108):


The steepness of the curve varies from point to point. At every point it has a definite numerical value, though, just as your automobile has a definite speed at any point while you are accelerating--namely, the speed you see if you glance at the speedometer.

That made me realize how naively we move from a mathematical model to reality. Does reality really have "points"?

Zeno made the assumption that reality did have points and from there concluded that no motion could occur. I think Zeno was right. If space and time were mathematical lines with points, nothing could happen. I don't know if Zeno thought his conclusion implied that no motion really occurred or whether he was trying to show that reality could not be made out of mathematical points.

It was not just philosophers who questioned mathematically continuous reality. Quantum physics started with rejecting a view that energy was equitably distributed across an infinite number of frequencies in the black body problem. Planck got around the problem by saying that energy was quantized and not continuous. Then things started to work.

Given Zeno and Planck, it is safe to say that reality and a mathematical continuum need not mix except at a level of approximation where one isn't looking too closely.

[Dreamwoven]

It was when the discussion became mathematical that I lost the thread.

[YesNo]

Cosmology today is full of mathematics. Rather than having conscious Gods who act as agents, like ourselves, such as Zeus or Thor or Brahma or Yahweh we have today unconscious mathematical equations with t representing time that has been objectified into a line of points.

Mathematics is a god with a lower "g" because that god is unconscious. But we don't think of these modern cosmological stories as myths and literature because we are caught in their enchantment, or bedevilment, depending on one's perspective.

[desiresjab]

I picked up Prime Obsession a few months ago and have been proceeding by jerks and starts until I am about 3/4 through. It always helps to get more things straight about these topics that are so difficult one may not approach them directly. One learns never to be surprised to find that work by Euler led to major developments by Reimann, or that Poincare anticipated giants not only in relativity but fractal geometry as well, which is a first cousin of chaos theory. Like Reimann, Poincare possessed high gifts in both math and physics. But he got to live longer. He was also a communicator whereas Reimann is often described as painfully shy.

Anyway, I know nothing of Bergson. From reading what you wrote I am unsure if his work was mainly philosophical or whether he muddied his hands with mathematics. I could check Wiki-pejia.

I am wondering how much impact a philosophical interpretation can have. It can alter the administration of things but cannot alter any mathematical truths. I believe philosophical interpretations are very important to the development of theories nonetheless. They can nudge toward particular investigations if the philosophical theory happens to be correct.

Anything that might reduce the time involved for grappling with old arguments is all right with me. As we discussed earlier and now again, issues from 1916 and 1922 are still being seriously debated in physics.

Mathematics, philosophy and physics are thoroughly bound up with each other now, because so many discoveries in math and physics bring heavy baggage to the table and renew the call for philosophy. Popularizers are indespensable for 99.99999% of people. Only the ones able to dig all the way to the roots are technically self reliant. Those are the people like Gauss, Einstein, Reimann, Poincare and Tao et al, and their bright disciples.

The long and short of that is, unless one is extremely bright and undertook this journey from an early age, acquiring all the right tools to investigate relativity or the Reimann conjecture, one's understanding will be a popularized understanding bereft of the technical details required for a truer grip.

Philosophy itself is changing as the need to interpret an explosion of physical theories presents new challenges. Technically versed writers operating one or two levels of abstraction above the "machine language" of folk like the prestigious list above, interpret the "message" for the interested masses, who are yet several more levels of abstraction above. Though not a new face of philosophy, it is now necessarily a prominent one.

It seems my vacation has made me talkative.

Einstein was a great image. The wild hair alone set him apart. His scientific disputes were more gentlemanly than most. But let us not forget, it was Einstein and not the others who made a verifiable prediction on the transit of Mercury. Predictions verifiable through measurement or experiment are of extreme value in scientific accomplishment.

The above merely by way of offering an explanation of why Einstein's interpretation of space-time might have prevailed philosophically over his anticipators and rivals.

All things are politicized, especially in our era. You can't get a right answer to anything, and that is typical.

[YesNo]

I am about half way through Prime Obsession and it is putting the Riemann Hypothesis in a perspective, both mathematical and historical, that makes sense. I'm glad to be reading it.

After reading Canales I don't think spacetime is anything more than a fiction. I don't know how the transit of Mercury fits into the creation of the brand name "Einstein", but I see "Einstein" as a marketing brand for an underlying cultural commodity. I don't think reality can be divided into points or instants. Planck's constant would be one argument against that and then there are Zeno's arguments that such a view would not allow motion to exist.

[desiresjab]

I am about half way through Prime Obsession and it is putting the Riemann Hypothesis in a perspective, both mathematical and historical, that makes sense. I'm glad to be reading it.

After reading Canales I don't think spacetime is anything more than a fiction. I don't know how the transit of Mercury fits into the creation of the brand name "Einstein", but I see "Einstein" as a marketing brand for an underlying cultural commodity. I don't think reality can be divided into points or instants. Planck's constant would be one argument against that and then there are Zeno's arguments that such a view would not allow motion to exist.

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

Anyway, once we come to terms that mathematics is not one of these fictions, we are able to make a necessary separation. Mathematicians themselves proceed creatively, but the end result is the discovery of the obvious, the discovery of the only way things could ever have been with regard to the numbers.

The numbers can cause scientific theories to bloom or fade. An equals sign means eqaul cardinality, in the end. Whenever, finally, the numbers do not work out, the theory does not work out either, and goes away. String theory may be in the process of doing this.

Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

Up to a point somtimes quite deep mathematics will support wonderful fictions then suddenly leave off its protection, exposing the fiction as a false route out of the maze. Euler once devised a formula that explicitly produced someting like the first forty-some-odd primes but thereafter could no longer be trusted.

Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

[desiresjab]

Since I am having a lot of thoughts right now, I may as well go on with them. With regards to my conjecture involving twin primes, its extension to prime triplets is put in serious jeopardy by the simple observation that two 4n+3 primes that are almost unfathomably far out the number line and only four units apart, will produce something extraordiarily close to a square yet which a diagonal will neverhteless always cut into two unequal quantities of lattice points in the appropriate quadrant of Eisenstein's diagram. This closeness to a square is the main reason I made the conjecture in the first place, so in some sense that puts the whole conjecture at jeopardy. I need a way to attack the problem.

[Dreamwoven]

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

Anyway, once we come to terms that mathematics is not one of these fictions, we are able to make a necessary separation. Mathematicians themselves proceed creatively, but the end result is the discovery of the obvious, the discovery of the only way things could ever have been with regard to the numbers.

The numbers can cause scientific theories to bloom or fade. An equals sign means eqaul cardinality, in the end. Whenever, finally, the numbers do not work out, the theory does not work out either, and goes away. String theory may be in the process of doing this.

Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

Up to a point somtimes quite deep mathematics will support wonderful fictions then suddenly leave off its protection, exposing the fiction as a false route out of the maze. Euler once devised a formula that explicitly produced someting like the first forty-some-odd primes but thereafter could no longer be trusted.

Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

I basically agree with desiresjab on this.

[YesNo]

Since I am having a lot of thoughts right now, I may as well go on with them. With regards to my conjecture involving twin primes, its extension to prime triplets is put in serious jeopardy by the simple observation that two 4n+3 primes that are almost unfathomably far out the number line and only four units apart, will produce something extraordiarily close to a square yet which a diagonal will neverhteless always cut into two unequal quantities of lattice points in the appropriate quadrant of Eisenstein's diagram. This closeness to a square is the main reason I made the conjecture in the first place, so in some sense that puts the whole conjecture at jeopardy. I need a way to attack the problem.

I think you might be right about the twin primes and the number of lattice points, but all I have to go on are tests for twins below 100. I don't understand what you are saying about prime triplets. They would be numbers of the forms: p, p + 2 and p + 6 or p - 4, p and p + 2 where all of these are prime.

[YesNo]

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

The problem is that mathematics need not apply to the universe. I view mathematics as a game that might have some practical value, but need not have any practical value. I see mathematics like the game of chess. Although chess has kings, queens, knights, bishops and pawns, we cannot expect those fictions to represent real kings, queens, knights, bishops or peasants any more than we can expect a mathematical structure to represent reality.



Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

The universe has to be finite for life to exist in it. That would be suggested by Olber's paradox. Now there may be infinitely many universes and I suspect there are other universes than ours given the evidence that ours had a beginning, but outside of that possible infinity of universes, transfinite numbers have no use value.



Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

I am not saying that numbers are not useful. All I am saying is that the universe does not go arbitrarily small which would be required if points actually existed. The physical justification for that is the need for Planck's constant.



Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

The problem with mathematics is that we think its ability to perform an analysis step such as splitting composites into smaller primes and then doing a synthesis step of multiplying those primes to get the composite back again is something that might also work in physical reality. It might not. That is, mathematical reductionism, represented by the reduction of composites to primes, may only work well within mathematics.

[desiresjab]

I think you might be right about the twin primes and the number of lattice points, but all I have to go on are tests for twins below 100. I don't understand what you are saying about prime triplets. They would be numbers of the forms: p, p + 2 and p + 6 or p - 4, p and p + 2 where all of these are prime.

What I now realize is that tremendous size and lying close together is not the key at all, is not what makes two primes behave a certain way in QR. The key is just as simple, however. The key is how many factors of 2 are involved in (p-1))(q-1), from the pure Eisensteinian perspective.

With two 4n+3 primes, p-1 and q-1 each have only a single factor of 2 to contribute. That is to say after dividing the total number of interior lattice points by 4, we come to an odd number, which of course cannot be divided evenly, so the two numbers have to have opposite characters when WAXY is divided once more by the diagonal. If only one more factor of 2 is available (which it always will be, as long as both primes are not 4n+3) to deal with this further division performed by the diagonal, then the diagonal will be dividing (apportioning) an even number of points in WAXY. If the power of 2 in the multiplication is only 23, then WAXY is forced to produce negative exponents for both primes upon the further division by the diagonal. But if the power of 2 is 24 or greater the exponents must always both be positive.

It appears that the nature of the exponents (and thereby reciprocity) depends only on the power of 2 in (p-1)(q-1), nothing else, in terms of Eisenstein's representation. The essence, the causal mechanism is none other than highly evenness. This was my original insight when I first started thinking about Brocard's problem and switched to QR, and before I actually understood what I was talking about. It has taken me this long to understand that my own insight was hitting the nail on the head squarely, to intend a pun.

With twin primes we are always guaranteed at least 23. What we simply need in order to always be even, i.e. in one another's residue set, is a 4n+1 number wherein n itself carries at least one factor of 2. From knowing nothing more we can always state the character of both primes with respect to each other in QR.

* * * * *

QR does not work when p=q. Imagine an 11X11 square. φ is 110, not 100. This means you cannot even get four squares (not rectangles in this case) all with equal lattice points.

But what about a 17X17 square where there are plenty of 2's to go around? This case will provide four equal squares all right. But it is a dead end, a non-sequitur, because no number between 1 and 16 inclusive will ever square out to 17 (mod 17), and so forth for all primes.

A visibly cogent fact is that the line p=q on graphing paper is a 45 degree angle and is our diagonal, and goes through all the points (1,1), (2,2), (3,3),...(17,17). The method does not work on squares. It only works on rectangles. The diagonal hits eight lattice points in WAXY. 256 interior lattice points divided by four is 64 for our quadrant square, but eight of these cannot count because they hit lattice points, bringing WAXY down to 56 servicable points, and each small triangle to 28, indeed equal, but meaningless except perhaps for why it is meaningless. Only on rectangles where p≠q are there no lattice points on the diagonal. P and q respond identically but meaninglessly when p=q because they do not kick against one another rationing out squares under the other as modulus. At the moment I do not know how to subtract those eight extra points in the context of something meaningful, I just know eight would have to be subtracted in this particular case to somehow fictionally redirect the apparently nonsensical. This is all about finding the logic of why it is illogical for squares themselves.

Only on rectangles where p≠q are there no lattice points on the diagonal.

So where p=q, it would have to look like:

[(p-1)(q-1)]-[(p-1)/2].

This is

p2-2p+1-(p-1)/2, is equivalent to

2p2-4p+2-p-1=2p2-5p+1, which means nothing to me but the sense of the nonsense.

A modulus is about division and remainders, and division is about ratios, and QR is about two unequal primes acting as units for the other under the operation of squaring, spitting out squares as remainders. Pitcher and catcher. Then switch places while the other acts as divider and see which numbers its overlap spits out as squares.

* * * * *

An interesting note:

In Prime Obsession Derbyshire states that 4n+3 primes consistently out number 4n+1 primes. There may be one brief interlude where 4n+1 primes hold the lead, but then it reverts back to a 4n+3 lead, supposedly for good. If they will always hold the lead is probably unproven. I cannot remember, or if he said.

[desiresjab]

Further note:

φ(p)-φ(p-1) seems to be the proper calculation for subtracing eight which I was trying to arrive at above for a 17X17 square.

[YesNo]

What I now realize is that tremendous size and lying close together is not the key at all, is not what makes two primes behave a certain way in QR. The key is just as simple, however. The key is how many factors of 2 are involved in (p-1))(q-1), from the pure Eisensteinian perspective.

With two 4n+3 primes, p-1 and q-1 each have only a single factor of 2 to contribute. That is to say after dividing the total number of interior lattice points by 4, we come to an odd number, which of course cannot be divided evenly, so the two numbers have to have opposite characters when WAXY is divided once more by the diagonal. If only one more factor of 2 is available (which it always will be, as long as both primes are not 4n+3) to deal with this further division performed by the diagonal, then the diagonal will be dividing (apportioning) an even number of points in WAXY. If the power of 2 in the multiplication is only 23, then WAXY is forced to produce negative exponents for both primes upon the further division by the diagonal. But if the power of 2 is 24 or greater the exponents must always both be positive.

It appears that the nature of the exponents (and thereby reciprocity) depends only on the power of 2 in (p-1)(q-1), nothing else, in terms of Eisenstein's representation. The essence, the causal mechanism is none other than highly evenness. This was my original insight when I first started thinking about Brocard's problem and switched to QR, and before I actually understood what I was talking about. It has taken me this long to understand that my own insight was hitting the nail on the head squarely, to intend a pun.

With twin primes we are always guaranteed at least 23. What we simply need in order to always be even, i.e. in one another's residue set, is a 4n+1 number wherein n itself carries at least one factor of 2. From knowing nothing more we can always state the character of both primes with respect to each other in QR.

It makes sense that the factors of 2 would be important here.




QR does not work when p=q. Imagine an 11X11 square. φ is 110, not 100. This means you cannot even get four squares (not rectangles in this case) all with equal lattice points.

But what about a 17X17 square where there are plenty of 2's to go around? This case will provide four equal squares all right. But it is a dead end, a non-sequitur, because no number between 1 and 16 inclusive will ever square out to 17 (mod 17), and so forth for all primes.

A visibly cogent fact is that the line p=q on graphing paper is a 45 degree angle and is our diagonal, and goes through all the points (1,1), (2,2), (3,3),...(17,17). The method does not work on squares. It only works on rectangles. The diagonal hits eight lattice points in WAXY. 256 interior lattice points divided by four is 64 for our quadrant square, but eight of these cannot count because they hit lattice points, bringing WAXY down to 56 servicable points, and each small triangle to 28, indeed equal, but meaningless except perhaps for why it is meaningless. Only on rectangles where p≠q are there no lattice points on the diagonal. P and q respond identically but meaninglessly when p=q because they do not kick against one another rationing out squares under the other as modulus. At the moment I do not know how to subtract those eight extra points in the context of something meaningful, I just know eight would have to be subtracted in this particular case to somehow fictionally redirect the apparently nonsensical. This is all about finding the logic of why it is illogical for squares themselves.

Only on rectangles where p≠q are there no lattice points on the diagonal.

So where p=q, it would have to look like:

[(p-1)(q-1)]-[(p-1)/2].

This is

p2-2p+1-(p-1)/2, is equivalent to

2p2-4p+2-p-1=2p2-5p+1, which means nothing to me but the sense of the nonsense.

A modulus is about division and remainders, and division is about ratios, and QR is about two unequal primes acting as units for the other under the operation of squaring, spitting out squares as remainders. Pitcher and catcher. Then switch places while the other acts as divider and see which numbers its overlap spits out as squares.

The idea of two unequal primes acting as units makes sense.



An interesting note:

In Prime Obsession Derbyshire states that 4n+3 primes consistently out number 4n+1 primes. There may be one brief interlude where 4n+1 primes hold the lead, but then it reverts back to a 4n+3 lead, supposedly for good. If they will always hold the lead is probably unproven. I cannot remember, or if he said.

I think I remember reading something like that in Derbyshire's text. I think ultimately the ratio of the number of primes in the two sets are supposed to converge to 1 implying they have the same number, but initially the 4n + 3 set has more. I couldn't find the page to reference it.

[desiresjab]

It just so happens it would be possible to devise extremely far from square rectangles in which either p-1 or q-1 was loaded with factors of 2. We assume p and q are both odd primes, as usual, with one of them a 4n+1 prime and the other a 4n+3, to keep matters clear, and to put the burden of factors of 2 entirely on the minus one of the 4n+1 prime.

It takes two dvisions by 2 to divide the rectangle into quadrants, and one more division by 2 to diagonally divide the bottom left and top right quadrants as in Eisenstein's diagram.

The p-1 or q-1 of all 4n+3 primes have only one factor of 2 to give. As long as the 4n+1 prime provides two factors of 2, which it must at least do by its nature, all three divisons can take place preserving the possibility of triangles WAY and YAX containing the same number of lattice points.

We do not really expect it, though, in the case of a very eccentric rectangle ABCD. Quite the opposite. We expect there not to be the same number of lattice points in WAY and YAX.

My conjecture about twin primes seems to involve both concepts--low eccentricity and factors of 2. So far, I am too dumb to prove it. I cannot even say if it is provable, or if it has already been proven.

[desiresjab]

Circumstantial support for the conjecture also lies in the fact that in all cases where p has more than two factors of 2, the diagonal will be dividing columns of even numbers. If WAY and YAX differ in the number of their lattice points they must differ by at least two, which is harder geometrically to do for a low eccentricity rectangle than to differ by one, it seems to me. I don't see how it could happen. I say it cannot.

I can close in on it logically but so far cannot find a mathematical method to prove it.

[desiresjab]

Even numbers of course can be partitioned into two evens or two odds in a variety of ways. 12 is 11+1 and 10+2, also 9+3 and 8+4, etc.

To acheive an eccentric additive partition in an Eisenstein count of lattice points would seem to requires high eccentricity on the part of the rectangle ABCD, i.e., less squareness. Right?

The eccentricity is why for the eccentric Eisenstein rectangle 5X13 (two 4n+1 primes), the count for WAXY is 12 lattice points yet a division into YAX and WAY through a diagonal division yields 7+5, allowing the QR to be negative, since 5 and 7 are both odd, all that is required.

The above result begs the question of whether there are two 4n+1 primes vastly far out the number line which are only four units apart, giving their rectangle profoundly low eccentricity, yet which somehow acheive a difference of two in their count of lattice points for YAX and WAY? Or are they all forced due to extremely low eccentricity to always be even, even, and furthermore identical in value?

This is another perspective to the conjecture.

[desiresjab]

A key thing to consider is that if you do choose two 4n+1 primes far out the number line separated by only four units, one of them is forced to have only two factors of 2. Evennes follows a pattern in the even numbers. Here is a list read left to right for the even numbers 2, 4, 6, 8...., showing how many factors of two each even number contains.

1, 2, 1, 3, 1, 2, 1, 4 1, 2, 1, 3, 1, 2, 1, 5 1, 2, 1, 3, 1, 2, 1, 4

1, 2, 1, 3, 1, 2, 1, 6 1, 2, 1, 3, 1, 2, 1, 4 1, 2, 1, 3, 1, 2, 1, 5 ...


My guess is that if the minus ones of two "twin" 4n+1 primes far out the number line can indeed acheive an eccentric partition (anything other than a split down the middle) it must be because as we can see from the list, that one of them has a mere two factors of 2. The other one can be slightly richer in 2's, or vastly richer. Now I am wondering if the size of that difference plays any part, if, of course, an eccentric partition under any condition can occur at all.

I don't think anyone else is going to do it, so it must be up to me. Surely there are a few sets of twin 4n+1's far enough out for low eccentricity yet within range of calculational investigation using some available tools.

The list above is called the ruler function. Believe it or not, that discontinuous thing has even been re-tooled for calculus.

[desiresjab]

To get off the math, Does anyone believe there is such a thing as the Nature of Reality, or is that another comfortable metaphor?

Is what we perceive reality? Does this include people hallucinating, too?

Or in the night imagining some fear
How easy is a bush supposed a bear
--Shakespeare

How much is objective reality, and how much is subjective reality?

Whatever flames upon the night
Man's own resinous heart has fed
--Yeats

[Dreamwoven]

I am much more at home in the fuzzy world of reality than in the artificial world of statistical "accuracy".

[YesNo]

It just so happens it would be possible to devise extremely far from square rectangles in which either p-1 or q-1 was loaded with factors of 2. We assume p and q are both odd primes, as usual, with one of them a 4n+1 prime and the other a 4n+3, to keep matters clear, and to put the burden of factors of 2 entirely on the minus one of the 4n+1 prime.

It takes two dvisions by 2 to divide the rectangle into quadrants, and one more division by 2 to diagonally divide the bottom left and top right quadrants as in Eisenstein's diagram.

The p-1 or q-1 of all 4n+3 primes have only one factor of 2 to give. As long as the 4n+1 prime provides two factors of 2, which it must at least do by its nature, all three divisons can take place preserving the possibility of triangles WAY and YAX containing the same number of lattice points.

We do not really expect it, though, in the case of a very eccentric rectangle ABCD. Quite the opposite. We expect there not to be the same number of lattice points in WAY and YAX.

My conjecture about twin primes seems to involve both concepts--low eccentricity and factors of 2. So far, I am too dumb to prove it. I cannot even say if it is provable, or if it has already been proven.

Prior to proving the conjecture, just stating it is important.

There appears to be more to the conjecture than that twin primes have the same number of lattice points in Eisenstein's diagram. How would you state your conjecture more generally? Are there other pairs of primes, beside twins, that should have the same number of lattice points as twin primes would? Could I assume pairs of primes, one of the form 4n + 1 and the other of the form 4m + 3 have the same number of lattice points?

[YesNo]

To get off the math, Does anyone believe there is such a thing as the Nature of Reality, or is that another comfortable metaphor?

Is what we perceive reality? Does this include people hallucinating, too?

Or in the night imagining some fear
How easy is a bush supposed a bear
--Shakespeare

How much is objective reality, and how much is subjective reality?

Whatever flames upon the night
Man's own resinous heart has fed
--Yeats

I agree with Dreamwoven that statistical accuracy seems artificial. That even goes for the probability in the quantum wave equations. All that statistics does is makes things look objective, but that is because it is no longer talking about the particular reality that is in front of us.

Part of the problem of reality is that we think and hope it is unconscious and objective. However, everything, including quantum reality, may be participating in various forms of subjectivity that seem foreign to our own. There may be nothing that does not participate in some form of subjectivity.

It is convenient to find ways to objectify reality. Some people rely on sacred texts. Some people rely on mathematics. Both of these work to some extent in the sense that they justify a belief that reality isn't totally chaotic.

Does the indeterminism of quantum reality imply that quantum reality shares in a form of subjectivity of its own since it appears to be making choices when we ask it questions? I think it does. What difference does that make? Probably not much. We will still be looking for patterns we can rely on. We will still be looking to better understand the universe around us. The only difference is we should have a greater respect for the reality we participate in. It is not dead. It is not objective and it cannot be completely objectified through statistics, mathematics or a set of sacred texts.

[desiresjab]

I agree with Dreamwoven that statistical accuracy seems artificial. That even goes for the probability in the quantum wave equations. All that statistics does is makes things look objective, but that is because it is no longer talking about the particular reality that is in front of us.

Part of the problem of reality is that we think and hope it is unconscious and objective. However, everything, including quantum reality, may be participating in various forms of subjectivity that seem foreign to our own. There may be nothing that does not participate in some form of subjectivity.

It takes a large pair of "mays" for what I have highlighted in red, mister.

Why must people complain that mathematical accuracy seems artificial, rather than accept that currently if there is any road toward this kind of knowledge it begins and ends somewhere with a guy tabulating statistics, and that he is, after all, tabulating in a language we created from the nature of what cannot be otherwise?

The point is, either someone has a better approach, or they do not. Of course, what good is it to us right now if it is not objectively presented? I admit there may be better ways. Will those ways please step forward?

The nice thing is that only means will the right interpreter step forward, the right philosopher, at whatever stage we are stalled. Someone once said:

Psychology is a body of theory awaiting phenomena; parapsychology is a body of phenomena awaiting theory.

Do not expect numbers to write an equation for the Sermon on the Mount or the best way of life. But if someday a deep theorem paves the way to verifiable astral travel, do not be surprised either.


Does the indeterminism of quantum reality imply that quantum reality shares in a form of subjectivity of its own since it appears to be making choices when we ask it questions?

As Socrates did of those who leaned too heavily toward or against the existence of gods, I would call the above presumptuous at this stage, in fact throughout, from imply to of its own to making choices.

I feel imply is far too strong, it means something will follow or is a necessary extension or consequence of. At any rate, slang usuage of the word is not fit for philosophical discourse where precision is needed, I am hopeful you will agree.

To say quantum reality participates in a subjectivity of its own is open to literally anything, you must admit. I propose that subjectivity in the quantum world is statistics, and an objective grappling with what we can only call randomness looks very different when "experienced" from the quantum perspective.

Choice is a very interpretive choice of words when talking about electrons and their ilk. I think it is a purely semantic convenience.

[YesNo]

It takes a large pair of "mays" for what I have highlighted in red, mister.

I use "may" to not sound too dogmatic. Replace it with "is" if you like.



The point is, either someone has a better approach, or they do not. Of course, what good is it to us right now if it is not objectively presented? I admit there may be better ways. Will those ways please step forward?

For most people, for their immediate problems, a better approach is through some form of meditation which involves their subjectivity.

Edit: It occurred to me that another way, another better approach to knowledge that matters, is through "middle-way" ethics. Again this is a subjective and not an objective approach to knowledge. These approaches, meditation and ethics, subjective thought and intentional action, are not available to deterministic reality (computers, robots, etc) nor to any hypothetical, random, unconscious reality such as zombies.



Do not expect numbers to write an equation for the Sermon on the Mount or the best way of life. But if someday a deep theorem paves the way to verifiable astral travel, do not be surprised either.

I don't expect numbers to do that. That would be one reason why numbers are inadequate to answer most problems people actually have.



As Socrates did of those who leaned too heavily toward or against the existence of gods, I would call the above presumptuous at this stage, in fact throughout, from imply to of its own to making choices.

I feel imply is far too strong, it means something will follow or is a necessary extension or consequence of. At any rate, slang usuage of the word is not fit for philosophical discourse where precision is needed, I am hopeful you will agree.

To say quantum reality participates in a subjectivity of its own is open to literally anything, you must admit. I propose that subjectivity in the quantum world is statistics, and an objective grappling with what we can only call randomness looks very different when "experienced" from the quantum perspective.

Choice is a very interpretive choice of words when talking about electrons and their ilk. I think it is a purely semantic convenience.

My use of "choice" and "subjectivity", especially when discussing reality that seems very different from our own, requires definitions which can be accepted or rejected. I accept the following definitions.

If we test something and it gives an answer that is neither deterministic nor random, I define that answer as a "choice".

If we detect a choice, I define whatever reality made that choice as having a "subjectivity" allowing the choice to be made.

These definitions of "choice" and "subjectivity" could be applied to our own choices and subjectivity which is why I use those specific words and do not make up other words.

[desiresjab]

I am interested in seeing what has convinced you of quantum consciousness. I cannot read everything I would like to--I simply write too much--so I sometimes must rely on shortened syntheses from trusted associates. I do not know the details of these particular experiments, though I am familiar with the rudiments of wave-particle-slits experimentation. How did they set the experiment up in a way that led to your convictions, if that is not too strong a word?

[YesNo]

I don't know much about quantum physics, but I did try to understand it when discussing "many worlds" a few years ago on these forums. However, I don't think one has to know much about it to get the relevant points.

The key to the consciousness idea is "indeterminacy" which implies both that something is not completely determined and also not random as a coin toss. I'll admit that is my idea. Most people I've read who talk about quantum consciousness are referring to the consciousness of the observer, not what is observed.

In addition I am interested in the non-individualistic nature of these quantum particles and the non-local behavior of particles that are entangled. This assumes that space and time are determined by local properties that Einstein posited. Until I read Canales, I assumed they were true. Now I'm not sure.

The double slit experiments that impress me are those showing what happens on the detection screen when one or two slits are open. Those form the base cases. It makes it look as if the slits are determining what happens. Then one tries to know which slit a quantum particle actually went through. However, just knowing that changes the wave pattern on the detection screen making it look as if two single slits were used and not a double slit, but that is after the fact of having gone through the double slit and not two single slits. So passing through a double slit is not all that is affecting what the result on the detection screen will be. The non-individualistic nature of the process is seen by passing the quantum particles one by one through the double slit without knowing which slit they went through. The pattern then becomes the same wave pattern on the detection screen which is not a random pattern implying that each individual particle (if one can continue thinking of them in this way) went through both slits at the same time and interfered with itself.

There is an additional question of just what is it that is going through those slits and arriving at the detection screen. Is it worth continuing to use the "particle" metaphor if the particle is required to go through both slits at the same time and interfere with itself? Is it even worth calling it a "wave" since one can detect which slit it went through and break the wave pattern.

[desiresjab]

The key to the consciousness idea is "indeterminacy" which implies both that something is not completely determined and also not random as a coin toss. I'll admit that is my idea. Most people I've read who talk about quantum consciousness are referring to the consciousness of the observer, not what is observed.

It harkens back to "cosmic consciousness."



In addition I am interested in the non-individualistic nature of these quantum particles and the non-local behavior of particles that are entangled. This assumes that space and time are determined by local properties that Einstein posited. Until I read Canales, I assumed they were true. Now I'm not sure.

Entangled particles seem determined. The state of one can always be known by checking the other.


The double slit experiments that impress me are those showing what happens on the detection screen when one or two slits are open. Those form the base cases. It makes it look as if the slits are determining what happens. Then one tries to know which slit a quantum particle actually went through. However, just knowing that changes the wave pattern on the detection screen making it look as if two single slits were used and not a double slit, but that is after the fact of having gone through the double slit and not two single slits. So passing through a double slit is not all that is affecting what the result on the detection screen will be. The non-individualistic nature of the process is seen by passing the quantum particles one by one through the double slit without knowing which slit they went through. The pattern then becomes the same wave pattern on the detection screen which is not a random pattern implying that each individual particle (if one can continue thinking of them in this way) went through both slits at the same time and interfered with itself.

Sure, that is all very impressive and mysterious. It means there is an awful lot we cannot explain. But nothing in it comples me to believe quantum particles are conscious.

We have an outside light. On top of it is a sensor which detects light or darkness to determine when to turn the light on. Under your beliefs I would have to call the sensor conscious. When the sensor gets too covered in bird sh*t the light mistakenly stays on all the time. You would still call it conscioiusness, I guess. Is sensitivity to change consciousness?


There is an additional question of just what is it that is going through those slits and arriving at the detection screen. Is it worth continuing to use the "particle" metaphor if the particle is required to go through both slits at the same time and interfere with itself? Is it even worth calling it a "wave" since one can detect which slit it went through and break the wave pattern.

Wavicles.

[Dreamwoven]

This is a bit like the latest astronomy post on "multiverses". We can't confirm that our universe is the only universe, but we build theories on the assumption that it is the only one. You might like to look at that post from space.com.

[YesNo]

This is a bit like the latest astronomy post on "multiverses". We can't confirm that our universe is the only universe, but we build theories on the assumption that it is the only one. You might like to look at that post from space.com.

I couldn't find a specific article in the link. The idea of multiverses is confusing. I can think of three different versions of this.

1) Given that the big bang occurred, and the microwave background implies some beginning occurred, then this event should not be unique. That means other universes, like our own, exist.

2) Given that it is conceivable for cosmological "constants" to be such that life could not exist, the anthropic principle implies that other universes exist so that ours supporting life could have a random chance of being.

3) To avoid the indeterminacy at the quantum level, any possibility that appears indeterminate in our universe is realized in another universe that pops into existence as soon as the indeterminate event takes place.

Of these the first seems plausible. The other two are based on metaphysical need to avoid choices occurring.

[YesNo]

It harkens back to "cosmic consciousness."

That goes back to George Berkeley, I suspect. Although I don't have any problem with cosmic consciousness, I wonder if some form of consciousness is also at the quantum level to justify a panpsychism perspective.



Entangled particles seem determined. The state of one can always be known by checking the other.

The first one measured chooses the state for both. Indeterminacy does not mean there is complete freedom of choice. I also find the non-individualistic implications interesting.



Sure, that is all very impressive and mysterious. It means there is an awful lot we cannot explain. But nothing in it comples me to believe quantum particles are conscious.

I don't know that anyone else claims that this quantum reality is conscious. However, I don't usually have original ideas, so I suspect others have thought about it.

What compels me to consider that there is agency at that level is the absence of determinism and the absence of complete randomness. This makes me think that within some limited range a choice is made. That choice implies some form of consciousness.



We have an outside light. On top of it is a sensor which detects light or darkness to determine when to turn the light on. Under your beliefs I would have to call the sensor conscious. When the sensor gets too covered in bird sh*t the light mistakenly stays on all the time. You would still call it conscioiusness, I guess. Is sensitivity to change consciousness?

If one gets a lot of this quantum reality together one gets objects, some we've made, such as rocks and trees and sensors and computers. These objects are more predictable. I would not say that a computer is conscious because when it works is does so deterministically as a computer, not as the quantum reality that makes up that computer.



Wavicles.

One of the problems with giving something like this a name is that name makes it seem as if we know something about that reality when all we know is a metaphor which may be leading us astray. The waviness of the reality is only known from results on a detection screen. Hence we assume that some interference caused the pattern prior to getting to the detection screen.

[desiresjab]

I wonder if some form of consciousness is also at the quantum level to justify a panpsychism perspective.

You wonder? You certainly do. I would say you do more than that. This seems to be your primary philosophical obsession, to convince yourself or others that quantum particles make choices and the universe is conscious at many levels and therefore we have free will because of quantum indeterminancy, based on really no evidence except what you for some reason want desperately to believe. I think that is a religious doctrine, not physics or even philosophy, and you treat it like a religious doctrine. It is not a beginning point for investigation. There is no ineluctable truth contained that can be seen for certain, like there is with the simple statement two is the successor of one.


The first one measured chooses the state for both. Indeterminacy does not mean there is complete freedom of choice. I also find the non-individualistic implications interesting.

Mere words, my boy, mere words. You are demonstrating what happens when an individual (yourself) takes a philosophical model too seriously. You are against taking scientific and mathematical models too seriously, but apparently it is okay to do so with a philosophical one. Philosophical models (interpretations) are produced several levels of abstraction above the trench math and physics. They are quite simply loose interpretations and possibilities for the non technically inclined to consider of what may be occurring. These "explanations" are then simplified further and finally transferred upward to books for the general public. It isn't over though. They make it to this forum about eight levels of abstraction up from the real thing. Here, what gets said over and over with the thinnest of support is that electrons make choices. I do not see where this is the basis for a philosophy that one would keep going at it endlessly. Precisely, it is a wild theory, not an ineluctable interpretation.


I don't know that anyone else claims that this quantum reality is conscious.

All you do is claim it. I gave you a chance to convince me. Why could quantum reality not merely contain some consciousness rather be conscious, as you keep stating?

I live and sense in a mammal-sized reality. That does not mean to me that mammal-sized reality is conscious, it means I am conscious within it. Some boulders are mammal-sized.


What compels me to consider that there is agency at that level is the absence of determinism and the absence of complete randomness. This makes me think that within some limited range a choice is made. That choice implies some form of consciousness.

De-hynotize yourself, friend. Do not jump to the forking word choice at every opportunity. You have at best a vague suspicion, sir, a sneaking suspicion, as my mother used to say. ...the absence of determinism and the absence of complete randomness. With such statemnts you have hypnotized yourself by playing loosely with their meanings.There is random and there is non random. Random means you cannot make accurate predicitons of an outcome. The short and sweet of it...randomness is predictable only through luck, non randomness could be tracked down by a species with fine enough tools, i.e. there is a formula.

Randomness is a concept. The concept says there is no formula for this thing. If there does exist a formula for it, than it is not this thing.

You are operating under false assumptions, according to your own criteria. Randomness is a pure abstaction, it may not exist. It is one of those concepts like continuity of the number line and infinite divisibility. No one even knows if these concepts apply all the way down in nature, or if nature operates discretely at a wee level. Heat does, from Planck.

Here you are applying these abstract mathematical models (as you like to call them) to reality nonchalantly and saying you can form beliefs out of them. Can the universe at any level express complete randomness? Maybe not. It is only a concept, so far. Perhaps the universe can only asymptotically approach pure randomness. Even in that case, there should be a formula for the universe.

I am not surprised if pure randomness does not exist. Asymptotric randomness might come as close as any limit we set, however, or it might have its own limit in this particular (or any) universe, like the speed of light does.

The newly coined concept of asymptotic randomness still allows you that thin edge of non randomness you seek. Just remember, any non randomness is determinate. This non randomness you will define as choice

If we imagine a bell curve, high in the middle and approaching the axis asymptotically in each direction, I know that to the left the universe can at least approach pure randomness extremely closely.

Conversely, there is no obstruction abstractly at least to considering the other side of the graph where pure order is approached asymptotically. This might be the realm of heaven. It is an interesting notion but I am not forming any religious beliefs out of it myself.

Does more order, then, bring joy? There is plenty of order in a rest home and little joy. There has to be order and freedom.

[desiresjab]

No one should get in a tizzy over the statement that any degree of non randomness in our universal beginnings means theoretically our universe is reproducible with the right formula. For three centuries many great minds including Newton, Laplace and Poincare struggled with the mere three-body problem. The complexities of its families of solutions still dazzles is.

Imagine, then, trying to find a formula to derive the universe and its particles. Perhaps not impossible, but unimaginably close to impossible.

[YesNo]

There is no ineluctable truth contained that can be seen for certain, like there is with the simple statement two is the successor of one.

One of our differences is that I don't view mathematics as more than a game. You seem to think there is more to it.



De-hynotize yourself, friend. Do not jump to the forking word choice at every opportunity.

You'll have to do better than that to de-hypnotize me.



Randomness is a concept. The concept says there is no formula for this thing. If there does exist a formula for it, than it is not this thing.

The thing about determinism and randomness is that they are ways to avoid subjectivity and choice.

Since I don't see how we could even come up with the concepts of determinism or randomness without subjectivity and a choice to focus our attention on these concepts, our subjectivity and our ability to choose can't be reduced to these derivative concepts.

That would be another place where we disagree.

[desiresjab]

One of our differences is that I don't view mathematics as more than a game. You seem to think there is more to it.

You will call it a game, come tsunamis or asteroid hits, I know that. In that case I would at least like to hear form you that it is the game we found here, whose laws we did not invent but only formulated in more detail from what is abstractly necessary.

Which is it, my dear Yes/No? Did we wholly invent this game of mathematics, like we did chess, or did we find ineluctable rather than arbitrary rules were necessary even to do something as basic as counting properly in the universe we found ourselves in? You should be able to answer that without too much sophism about rules of a game.

1 The laws of Math are purely our inventions, just another game?

2 The laws of math are such that they must be as they are. We can discover and develop new byways in them but not invent the mechanics those laws rely upon.

Take a crack at it.

[YesNo]

Let's consider the claim that "two follows one". I agree with you that that statement is true in every universe since none of those universes care one way or the other about it. Similarly, the game Tic-Tac-Toe is true in any universe. A limerick I write is a limerick in any universe.

To get more specific, What does it mean to say that two follows one? If we are in the integers it means that there is a binary relationship, in this case a strict total ordering, containing the pair (2,1) requiring that 2 > 1.

We could easily switch that relationship around and say that "one follows two" by changing the binary order relationship to include (1,2) and not (2,1). In general if a > b then we make b > a. That is also true in every universe, because the existence of any of those universes is not dependent upon how we define that binary order relationship.

We can go further. Consider the finite field of two equivalence classes labelled 0 and 1. 0 contains all the integers that are even and 1 contains all the integers that are odd. In that field the statement "two, as an equivalence class, does not exist" is true in all universes.

None of those universes care how we define the rules of the game we are playing at the moment.

The answer to 1 is that mathematics is a lot like a story or poem. It objectifies what we subjectively understand. That objectification I can call a "game" although it could be a "story" or a "limerick".

The answer to 2 is similar. When I write a limerick it conforms to a certain pattern or it is not a limerick. When I construct a mathematical structure, what I derive from it follows logical patterns or it is not a mathematical structure.

The missing question is does the universe behave like any of these mathematical structures (or stories or poems or physical theories) that we might happen to create? We hope it does, but we are led into errors when we take these objective artifacts of our subjectivity too literally. In spite of what some people like to believe, we cannot completely dump our subjectivity into something objective.

There is another problem. When we do not have physical evidence for something we sometimes rely on these objectified, mathematical structures to guide us. This creates a blind spot. For example, is time in reality a mathematical continuum of infinitesimal points? We cannot physically verify infinitesimals given quantum limitations to the discrete. Furthermore, the assumption that time is a mathematical continuum leads to paradoxes in physical reality, notably Zeno's, even though the paradoxes might have been resolved mathematically.

[desiresjab]

Let's consider the claim that "two follows one". I agree with you that that statement is true in every universe since none of those universes care one way or the other about it. Similarly, the game Tic-Tac-Toe is true in any universe. A limerick I write is a limerick in any universe.

Tic-Tac-Toe is true in any universe, but it is not much of a tool for exploring universes. Its laws, too, were always possible in any universe, but it is not fundamental to any of them


To get more specific, What does it mean to say that two follows one? If we are in the integers it means that there is a binary relationship, in this case a strict total ordering, containing the pair (2,1) requiring that 2 > 1.

This argument does not describe a universe, friend, nor does it have anything to do with the cardianl successor of 1. You ask what it means. It means two is the cardinal successor of 1. Your argument is totally irrelevant, a true red herring.


We could easily switch that relationship around and say that "one follows two" by changing the binary order relationship to include (1,2) and not (2,1).

And you think this has anything to do with cardinal successors? This is as relevant as looking at 18 and claiming 8 must be the successor of 1.


In general if a > b then we make b > a. That is also true in every universe, because the existence of any of those universes is not dependent upon how we define that binary order relationship.

Again, sir, we are not talking about binary relationships. Notice you are not counting.


We can go further. Consider the finite field of two equivalence classes labelled 0 and 1. 0 contains all the integers that are even and 1 contains all the integers that are odd. In that field the statement "two, as an equivalence class, does not exist" is true in all universes.

Successors, sir, fundamental counting--that's where we are. You are not counting in this example, you are blinking two values as in QR.


None of those universes care how we define the rules of the game we are playing at the moment.

What makes you think so? They care a lot, if you want to put it that way. They care enough that they will not let you change any laws, you have to use the ones that preceded your arrival. The universe does not care about Tic-Tac-Toe because Tic-Tac-Toe is not integral to it.

A man made up Tic-Tac-Toe. Did a man make up the mechanics that causes 4n+1 numbers to behave differently than 4n+3 numbers in QR? A man saw that they behaved that way, a man did not make up that they behaved that way. They were behaving that way always, before Euler conjectured it and before Gauss proved it. A man did not make up the fact that numbers will behave this way. there is no other way possible for them to behave. Ineluctable: can be no other way.[/QUOTE]



The answer to 1 is that mathematics is a lot like a story or poem. It objectifies what we subjectively understand. That objectification I can call a "game" although it could be a "story" or a "limerick".

Before, you said mathematics was a stroy, now you say it is a lot like a story or a poem. Make up your mind.

You are really astray and I do not know if I can help you. You are cruising at a very high level of abstraction making judgements on the lowest level in the universe. Because a mailbox and a car are both red, at a particular level of abstraction they are alike. Because a peach and a baby are both soft, at a certain level of abstraction they are the same. As you descend to lower levels the differences betwen babies and peaches becomes clear. They are only superficially alike. You see there are a lot more differences than there are similarities between the two.

You are making high level abstractions to point out superficial commonalities.


The answer to 2 is similar. When I write a limerick it conforms to a certain pattern or it is not a limerick. When I construct a mathematical structure, what I derive from it follows logical patterns or it is not a mathematical structure.

High level of abstraction to point out superficial similarity.


The missing question is does the universe behave like any of these mathematical structures (or stories or poems or physical theories) that we might happen to create? We hope it does...

Which part of the universe are you talking about? Again, youapparently only mean cosmological or quantum scales. Here at mammal-scale, math works just fine to capture order and help us make better decision by the millions everyday.


There is another problem. When we do not have physical evidence for something we sometimes rely on these objectified, mathematical structures to guide us. This creates a blind spot. For example, is time in reality a mathematical continuum of infinitesimal points? We cannot physically verify infinitesimals given quantum limitations to the discrete. Furthermore, the assumption that time is a mathematical continuum leads to paradoxes in physical reality, notably Zeno's, even though the paradoxes might have been resolved mathematically.

You are like a madman that only knows certain things to repeat. There is no blind spot created by math. Infinitesimals cannot be verified. What are you going to do, find a smallest one?

Most of the mechanics of the universe were a blind spot to early man. Math reduces and illuminates them, it does not create them.Your problem is you are disappointed the lighting is not perfect. You are mad because when the light of mathematics is shined on the universe it does not show everything with 100% resolution. We have to know more laws than we do now. When those laws of numbers are found, some of them may apply to your obsession. Until then, hope for a great mathematician to arise, because the solution if it comes will be in mathematics, it will not come from people with prayer books. The people with prayer books are not even searching for the same kind of answer. I would consider it a miracle if people praying were ever of instrumental use in science.

[YesNo]

This argument does not describe a universe, friend, nor does it have anything to do with the cardianl successor of 1. You ask what it means. It means two is the cardinal successor of 1.

Where is the number 1 in the universe?

[desiresjab]

It is not anyplace.

[YesNo]

I see. It is a game we created. My cat doesn't play the game (to my knowledge). I don't even think my computer really knows what the number 1 is.

[desiresjab]

I see. It is a game we created. My cat doesn't play the game (to my knowledge). I don't even think my computer really knows what the number 1 is.

Am I still banned?

[desiresjab]

I don't know if you got the question out just right. But since it is the big question in all of this, I have to address it.

If we start with 1, it would not hurt to say what 1 is, right? I think that is what you meant.

Words like "unity" or "singularity" only stall the answer. They are not what a person who asks the question means to get at. Those two words already contain the notion of oneness, so we cannot use them to define oneness, can we now? I mean really.

Everything else is defined neatly in terms of 1. But what is 1? That is it, isn't it? Here is where it gets thorny philosohically.

Everything after 1 is mathematically defined, but is 1 mathematically defined?

Even though there is a mathematical operation 0+1=1, that would be getting ahead of ourselves.

The idea of 1 seems to be philosophical. Translated to counting, it can only be existence itself, as opposed to no existence. 1 and 0 are existence and non existence, mathematically. Ah! but that is one level of abstraction up from the abstract concept of existence itself, isn't it? I just learned something.

Further, the reason mathematicians did not start at 0 to define numbers is that they would then would have to explain existence itself when 1 appeared from nowhere.

1 is the assumption of existence, which is the foundational assumption of mathematics, maybe of philosophy too. Once we do that, everything follows nicely, in math at least, and we can even go back at that point and say, yes, 0+1=1.

Unless you must call the assumption of existence a game, where is the game? We conceive of nothingness, observe we exist, and further observe there is a proliferation of things that exist. If you want to enumerate things that exist accurately you must use numbers to count them. Each number is the successor of the number in front of it, exceeding it by exactly one, the unit already defined and brought out of nothingness to represent existence. One is one of anything—a car, a house, a dumpling, an idea.

We know we exist. I myself believe we are not what we think we are by a long shot, but in one way or another we and other things do exist. We have every right to assume that we exist and that other things exist as well.

We invented symbols for existence and non existence, 1 and 0. Man was well along in the civilization process before the idea of 0 came and stuck.

The truth is, historically mankind started with one, when it came to counting, since it was a reasonable assumption that things existed. It was a long time later when they got around to explaining how zero worked in every situation. Very little in math ever came easy after counting. Counting was pretty natural, in the sense that enumeration could easily be verified with the senses, but counting still took a long while to develop anyway.

I do not see counting as a game, nor the assumption of existence as making a rule. I do not see keeping track of enmueration a game.

Math can be used like a game, one can think creatively and inventively with it, but the fundamental propositions of mathematics are no game, they came right out of counting what was already there using number successors before anyone ever bothered with the concept of number successors.

[desiresjab]

It is OK to view nature as your opponent to be mastered, but you would not want to mistake any model for the actual, even the game model for mathemathics, right?

Math can be fun like a game. I want it to be. It is to many.

Even non commutivity in multiplication came about to satisfy multiplication of matrices and groups, not as an arbitrary experiment with operations. Matrices are shadowed by much in nature and our daily lives. Matrices are a brand of mathematics that does not count things, but the magnitudes of the numbers in the columns and rows of the matrix are interpreted exactly as in counting. 26 is still 26.

It is all right to approach mathematics like a game when you do math research. It should be fun. Satisfying your curiosity is a fun thing.

In order for matrices to reflect the way certain things happen, one operation (multiplication) had to be tweaked. Galois was following nature, maybe looking for it.

Since the whole dispute in my mind now comes down to the number 1 and how it got there, I can more easily make my points.

[YesNo]

I keep getting those "you are banned" messages when trying to post as it looks like you do as well.

If our subjectivity is not a game, then neither is mathematics. I would also elevate poetry to being something more than a game.

However, what comes out of our subjectivity is partial. We cannot take it too literally.

[desiresjab]

I keep getting those "you are banned" messages when trying to post as it looks like you do as well.

If our subjectivity is not a game, then neither is mathematics. I would also elevate poetry to being something more than a game.

However, what comes out of our subjectivity is partial. We cannot take it too literally.

Yeah. I consider normal language much more complex than mathematics. Programming Big Blue to beat Kasparov in chess was fantastic, but the far more formidable task was programming it to beat Ken Jennings at jeopardy. The procedures of chess are rather mathematical at heart, jeopardy is not. I believe Big Blue was not allowed to read the questions--for that would have happened instantaneously, but had to hear them and understand them. Big Blue had to understand all the puns and allusions in the typical jeopardy question. This is much closer to understanding poetry than it is math.

Connotation and suggestion is so complex. The same images will not form in our minds as we read Shakespeare, the same thoughts. Words do not have equal signs between them, even synonyms do not. Every word is different from every other. The same word will have different connotations in a different setting. This not true of the number 6.

[YesNo]

The number 6 can mean all kinds of things to us. It can be a composite number. It can be one-third of the "beast" 666. It can suggest a six-pack of whatever we want at the moment.

[Dreamwoven]

I keep getting those "you are banned" messages when trying to post as it looks like you do as well.

This is weird, that both of you have got such banned messages

[YesNo]

Although I probably deserve it for all my hell-bent sins, it seems that we get them when we use special characters to format math concepts in a post and the software thinks we are using a browser that does not allow ads to display.

[Dreamwoven]

What is a special character? Like Chinese? It would be nice to be rid of ads, if that is possible.

[YesNo]

Like the symbols above the numbers on the keyboard. I figured it is best not to touch them at least when there are numbers next to them.

As far as ads go, I usually don't mind them. I have been known to click on one or two. They just have to display rapidly.

[desiresjab]

Like the symbols above the numbers on the keyboard. I figured it is best not to touch them at least when there are numbers next to them.

As far as ads go, I usually don't mind them. I have been known to click on one or two. They just have to display rapidly.

I am pretty sure that is not it. The site was recently restructing some stuff.

[desiresjab]

Yes/No, even now I ponder quadratic reciprocity everyday. I love the concreteness of it compared to abstact philosophical talk.

Like I said before, I am dumb, so it takes me a long time to see things.

But by putting a little bit next to a little bit, I have finally seen what I wanted to see, not more than five minutes ago for the first time.

No abstract algebra or group theory needed, just a minute inspection of the details of Eisenstein's proof.

One can see and understand almost all the details of Eisenstein's proof without understanding why it proves QR.

I had already figured out that the dimensions of the rectangle represented the scale of the relative sizes of the moduli working aginst each other in QR. What I had not put into words was that this representation of scale is only activated by the diagonal. Hold that thought.

* * * * *

The other thing is a clear concept of just what the multiplication

[(p-1)/2] [(q-1/2)] stands for. What does it stand for? First, each is the number of respective quadratic residues of the primes individually.

This multiplication stands for any one of 5 things combined with anyone of 3 things. In other words, it counts how many ways the number of quadratic residues of each prime can be combined with one another, fifteen ways, in this case.

Each combination has its chance. Each combination is a lattice point. The diagonal slices WAXY one more time, dividing the number of lattice points a final time. If it is slicing through an odd number of lattice points, triangles WAY and YAX are forced to different parities. This only happens if they are both 4n+3 primes.

The diagonal expresses the gear size of each prime. Set with the bigger prime as width rather than height, I can get many more lattice points in WAY than YAX with a large enough size discrepancy between primes. I am pretty sure of that conjecture. It is the polar opposite of my other conjecture.

Yes, p and q are the individual gears, but the diagonal is them meshed together.

[desiresjab]

We start with a p by q rectangle. We fill in the lattice points. We divide the rectangle by two vertically, and divide it by two again horizontally.

Only now do we divide it by two once more with the diagonal, allowing the diagonal to be "last agent," as you might prefer.

With the construction of the rectangle, the gears sizes are set. With the construction of the diagonal, the gears are meshed together and running.

Each gear is a period, a modular cycle of remainders. When you combine two periods, you get a larger period, like a period of 77 for pq, before everything is back to where it started. The original marks on the two gears will again be aligned vertically with a stationary reference point.

The upper triangle WAY can "hog" lattice points, because the extreme "lean" of the diagonal forces lattice points into the upper triangle WAY in the left hand lower corner of the rectangle, but the best the lower triangle YAX can ever do is break even.

[YesNo]

No abstract algebra or group theory needed, just a minute inspection of the details of Eisenstein's proof.

I agree that a better understanding should not need those tools. They help to generalize and perhaps prove results.



This multiplication stands for any one of 5 things combined with anyone of 3 things. In other words, it counts how many ways the number of quadratic residues of each prime can be combined with one another, fifteen ways, in this case.

Do you have an example? I don't follow the 5 and 3 things.



Each combination has its chance. Each combination is a lattice point. The diagonal slices WAXY one more time, dividing the number of lattice points a final time. If it is slicing through an odd number of lattice points, triangles WAY and YAX are forced to different parities. This only happens if they are both 4n+3 primes.

I agree. You will only get an odd number if both primes have remainder 3 modulo 4.



The diagonal expresses the gear size of each prime. Set with the bigger prime as width rather than height, I can get many more lattice points in WAY than YAX with a large enough size discrepancy between primes. I am pretty sure of that conjecture. It is the polar opposite of my other conjecture.

Is there an upper bound on this discrepancy?



Yes, p and q are the individual gears, but the diagonal is them meshed together.

"Gears" is a nice metaphor. I had not thought of it like that before.

[desiresjab]

I agree that a better understanding should not need those tools. They help to generalize and perhaps prove results.



Do you have an example? I don't follow the 5 and 3 things.



I agree. You will only get an odd number if both primes have remainder 3 modulo 4.



Is there an upper bound on this discrepancy?



"Gears" is a nice metaphor. I had not thought of it like that before.

5 and 3 are (p-1)/2 and (q-1)/2 when p=7 and q=11. The simple multiplication is counting the ways three objects can combine with five objects in pairs. There are fifteen different pairs representing how p can pair with q and vice versa. At this point the ground level mechanics are gone and we are looking for something else. We only need to keep our tether line connected to where we started from so we can remember where we are.

[YesNo]

I see. These numbers will change depending on the primes involved.

[desiresjab]

Duh, I must be slow. I have to admit, I either forgot or never realized that a simple multiplication represents how many ways the objects from two sets can be paired.

In QR I think it is important that (p-1)/2(q-1)/2 represents that, not just some normal product as we usually think of a multiplication. Basic multiplications are combinatorial, if you enlarge your viewpoint slightly. It gives something deeper to explore. If I can reverse map each of the fifteen lattice points... another revelation might be near. Ahem! A mirage is likely, too.

[desiresjab]

At least 227 proofs of QR are known. Even the few I know of use a staggering array of techniques and math. There are proofs emanting from:

1 Modular Arithemtic
2 The Pythagorean Theorem
3 Abstract Algebra
4 Group Theory
5 Geometry
6 Combinatorics
7 Trigonometry
8 The Binomial Theorem
9 Class Field Theory
10 Calculus (real anaysis)
11 Calculus (complex analysis)
12 Euclidian Algorithm
13 The Chinese Remainder Theorem
14 Vectors

Additional fields or functions that I suspect proofs emanate from:

1 Euler's Totient Function
2 Game Theory
3 The Divisor Function
4 Eliptic curves
5 Modular Functions
6 Discrete Logarithms
7 Primitive Roots
8 Fermat's Little Theorem
9 Statistics

Each of the fields probably has produced numerous proofs with slightly different twists. QR is so centrally connected, as I keep mentioning, or these diverse fields would not all have relations with it.

[desiresjab]

Of course each lattice point only represents any old pair of quadratic residues (from anything that is said in the proof). My experimental idea is to replace each lattice point in WAXY with a specific pair of residues There may be a revealing way of matching each particular lattice point to every specific residue pair. How to match them is an intriguing question, which I am hoping will later become obvious, because that would mean there is a superior way of mapping point to pair. I think I will carry out the number crunching. More later.

[YesNo]

Maybe you need to study one of the other QR proofs to help stimulate new ideas?

[desiresjab]

The two sets of quadratic residues for p and q are: {1 4 2} {1 4 9 5 3}.

The first set actually represents the height of the rectangle, though p in the final result is represented by the lower half of WAXY, and q the upper half, and q represents the width.

(1, 1) is the first pair, and we will take those to be coordinates. That one is easy to assign a point. (1, 4) is next, and we take those to be coodinates, too, and so on and so forth.

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
* * * *
(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)

Only those pairs with an asterik have unique coordinates in WAXY, the other pairs simply reduce to duplicates. See a pattern? Remember, the first coordinate is reduced (mod 3), and the second is reduced (mod 5). And do not forget!! our first coordinates above are vertical coordinates, and the second coordinates are the horizontal. That is the opposite of what we are all used to from algebra where the horizontal x-coordinate is always the first element in the ordered pair and the vertical y-coordinate is the second element.

If I reversed the order of the digits in the ordered pairs, we still get eight ordered pairs with asteriks, as long as I reverse the moduli too.

Were eight asteriks a coincidence above, or will there always be exactly the same number of pairs with natural coordinates as there are lattice points in one of the triangles of WAXY?

Only more grinding will know.

[desiresjab]

Maybe you need to study one of the other QR proofs to help stimulate new ideas?

This is a new idea, lad. It takes quite a lot of labor to investigate one idea that is already deep in, as you can see from my last post, which dealt with only one pair of primes, 7 and 11. I am so scattered around I cannot do it all in one day. But an amateur is having fun.

[YesNo]

Were eight asteriks a coincidence above, or will there always be exactly the same number of pairs with natural coordinates as there are lattice points in one of the triangles of WAXY?


I like how you paired the residues. I hadn't thought of them in that way before.

[desiresjab]

I like how you paired the residues. I hadn't thought of them in that way before.

I hadn't either. Last night I did more numerical experiments. Those pairs that work directly as coordinates within WAXY without being reduced, and those that do not, always seem to partition the two halves indentically as the diagonal does, but I can no longer conjecture the larger half will always be the one expressible as coordinates. For 5 and 11, WAY and YAX contained 6 and 4 lattice points respectively, but only four expressions for lattice points out of the ten total. There goes that one. The expression went with neither the larger half nor the one on top. At least we know the one on top will always have as many or more points than the bottom, since it wil always get the point (1, 1). Just the lean of the diagonal because of our convention of always making the short side the vertical dimension gets that done. Because we can remember a few conventions and have wiki-pejia access with Eisenstein's rectangle, we can communicate more easily. Tug lines are good in deep water.

More might come of investigating residue pairs further. It could also be a deadend that dispalys a lot of not unlikely connections without helping to get deeper into the process. Both are true of any investigation, however, and since I do not see any other method that might possibly lead ahead right now, I will roll it around for a while, as usual, without doing any more work.

If I can figure out what the number of natural expressions represents, that might be used as another means of going deeper into the machine.

I suppose the thing to look at now is the distribution of the "natural points" within WAXY, besides (1, 1), which is always predictable, and see if they tell me anything new or are simple reflections of ideas I already know.

One thing jumps out--the pairs that make it will be 1-heavy, with both coordinates small. The 1 column and the 1 row should be more full than the others.

[YesNo]

The residue pairs seem like an interesting approach. I don't know enough about the topic to know if anyone else has looked into this. It may lead to some other unexpected results.

[desiresjab]

The residue pairs seem like an interesting approach. I don't know enough about the topic to know if anyone else has looked into this. It may lead to some other unexpected results.

There are many experts right now who could tell us exactly where pairing residues leads. I never even pretend I might out think them or dream up an approach no one else has tried. The smartest people in math are simply too smart. Questions we have to dig in for some numerical on, they solve accurately in their visions. It is probable that if this leads anywhere we will end up in the territory of other maths we know nothing about, speaking in our usual language.

All the "unnatural pairs" pairs we generate with our multiplication are in reality duplicates of the natural pairs. I feel strongly that the natural pairs have some other quality not possessed by the pairs that have no representation at all. I could almost put it into words right now, but they would be only words with no mathematical help yet. It is a matter of plugging both sets in somewhere and seeing the difference in their behavior.

Now wouldn't it be a shade of wonderful if it was something as simple as they are divided according to those which are residues two ways and those which are quadratic residues only one way? Now I just have to decide how I am going to decode it. But how does that work if the division is something like 6 and 4, as it is with primes 5 and 11? I am learning not to box myself in with conjectures.

For now we can feel confident assuming we know the pairs will always partition out the same, through the coordinates or through the diagonal division, but are unable to yet say why or how. Euler might be giggling at us right now.

* * * * *

People might wonder how we can spend so much time on quadratic reciprocity without being even dumber than we admit to, when college juniors and seniors take classes in number theory where it is presented and pass their tests. It was just one more class in a long string of difficult classes for college students majoring in math or science.

Do they really understand it? They pass the tests, I believe we would pass them, too. Furthermore, we would amaze many, probably including the professor, with how thoroughly we understood many aspects of the theory.

I took calculus and differential equations, and I did better than merely pass those classes. What you learn in such classes is how to operate the formulae and which ones to use when. Normal college classes are not about in depth looks at particular problems, but learning each new language and how to operate in it. None of them go away with a greater understanding than we have right now.

As a graduate student in math they might confront this problem full on in a class, once they have had both group theory and abstract algebra. Those proofs are from the eagle's perspective, many levels above our ground level and subterranean approaches. Their view tells them something like the crankshaft will not turn, at any rate, if the carbeurator is disconnected. It does not take them down in the engine where the numbers are. At best, they learn how certain classes or fields behave, which is where we may end up yet.

The 38 lectures I watched on abstract algebra did not prove QR. A few times it seemed like they were getting close to the same ideas, though. They did prove quite a few other propositions, however, including Fermat's little theorem. The thing that knocks you off your feet is how brief the proofs are. A few sweeps of the chalk and they are done. That is how high they are soaring.

[desiresjab]

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
* * * *
(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)


Here are the quadratic pairs for 7 and 11. On this word processor, the asteriks may not line up where I want them. They didn't. The point is, if you take the moduli back to three and five, those fifteen points reduce back to only eight points. Only eight points are represented above. There are more lattice points in WAXY, that have no representation at all. Let us track down the other seven.

(1, 2) (2, 2) (3, 3) (3, 2) (4, 2) (3, 1) (3, 4)

the point (4, 2) is not actually in WAXY. It is too large. Reduced all the way it is merely the point (1, 2)

Wait again. (3, 5) is a point reperesented nowhere. (4, 2) can be reduced by the smaller moduli, (3, 5), cannot, except to (0, 0), which mamma don't allow none of around here.

(3, 5) must replace (4, 2) in the list of pairs above. Here it is:

(1, 2) (2, 2) (3, 3) (3, 2) (3, 5) (3, 1) (3, 4)

We have their names, now what can we investigate or interpret?

What can they not do that the others can?

[desiresjab]

* * * *
(1, 1) (1, 4) (1, 9) (1, 5) (1, 3)

(4, 1) (4, 4) (4, 9) (4, 5) (4, 3)
* * * *
(2, 1) (2, 4) (2, 9) (2, 5) (2, 3)

Only eight distinct points are represented above, if you use the reduced moduli. All are valid, they just happen to lie outside of WAXY on the coodinate system, though of course they are within the larger ABCD, to keep things in perspective. Magically, they all reduce back to natural pairs already listed within the matrix. There are more lattice points in WAXY. Let us track down the other seven.

(1, 2) (2, 2) (3, 3) (3, 2) (3, 5) (3, 1) (3, 4)

Here are the two residue sets again. {1 4 2} {1 4 9 5 3}.

We intuitively understand why the point (1, 1) is represented. Can we understand why a point like (1, 2) above is not? We want to understand it in a better way than just that the two sets when combined in the order shown, cannot produce that pair. What do we need to try?

For (1, 2)
1 is a residue of 7, but 2 is not a residue of 11. That is a start. Keep going.
For (2, 2)
2 is a residue of 7, but 2 is not a residue of 11.
For (3, 3)
3 is not a residue of 7, but 3 is a residue of 11
For (3, 2)
3 is not a residue of 7, and 2 is not a residue of 11
For (3, 5)
3 is not a residue of 7, but 5 is a residue of 11
For (3, 1)
3 is not a residue of 7, but 1 is a residue of 11
For (3, 4)
3 is not a residue of 7, but 4 is a residue of 11

Remember, these are pairs which are coordinates, but were not generated in our combinatorial multiplication. There is at least one-way rejection, for the entire list. But for the matrix of pairs above with fifteen pairs, both never reject, all the time. Even the pairs that lie outside of WAXY are valid, they are just not inside WAXY, which is true for some of the quadratic residue pairs generated in the multiplication. Since some of the valid pairs generated in the combinatorial multiplication lie outside of WAXY, we can only expect some of the points inside WAXY to be valid residue pairs. In fact, eight of them are, and seven of them are not. This accords exactly with the split of the diagonal, and the split of the natural expressions.

Those lattice points within WAXY that aren't two-way accepting, fill the second column and fifth rows, exclusively, forming a right-heavy bar on the capital T of its shape. This puts 5 of the 7 points in WAY, but 2 of them in YAX. Those seven are the same ones we said earlier had no “Natural expression.” Natural expressions accompany pairs of two-way acceptance only. We wonder on the side if only one is usual.

For primes 7 and 11 and the rectangle WAXY, eight of the lattice point coordinates (the same ones with natural expressions) do indeed have two-way acceptance pairs as coordinates, and seven have at least one rejection. A perhaps complexifying aspect here is that one of those seven pairs (3, 2) is two-way rejecting.

When you represent the pairs as coordinates when Eisenstein's rectangle ABCD is on coordinates, then eight of the fifteen pairs of coordinates within WAXY are coordinates where both elements are residues of their respective prime in the ordered pair. Those are exactly the same pairs that have natural expressions that were generated in the combinatorial multiplication. Six pairs of coordinates have only one quadratic residue, and one pair (3, 2), has none, i.e. 3 is not a residue of 7 and 2 is not a residue of 11. Graph-wise, (3, 2) is where the leg of the T and its bar intersect and occupy the same lattice point.

Somehow, the diagonal and the value of the number of natural expressions, manage to cut WAXY correctly in terms of two numbers to be used as exponents, innocent lattice points, and actual coordinates. The diagonal does not cut all the naturals to one side, but gets the numerical partition correct. The T marks the exact positions. That coordinates even apply is great!

7 and 11, being a pair of 4n+3 primes, are never mutal hosts to one another, but some of the pairs generated are, in a sense. We generated the pairs the old fashioned way—through a multiplication process more basic than the one taught in grade school, and found out that 15 is the number of distinct pairs generated from two sets of 3 and 5 members, respectively. Eight of these pairs are mutally accepting pairs, and seven of them are not.

Now it's back to the think tank.

[YesNo]

As you mentioned math classes are more to show students how to use the language and solve some problems. Generalizing does take one away from the details.

I was looking at the Sierpinski problem recently since that is the one that I have a computer working on for PrimeGrid. I am trying to see if I can get Python and MySQL to help generate covering congruences for some numbers that are unknown whether they are Sierpinski numbers or not.

A Sierpinski number is a number of the form k 2n + 1 where k is odd. A Sierpinski number is a k such that for all n none of the numbers are prime.

[desiresjab]

Paul Erdos made some discoveries concerning covering numbers. The Sierpinski gasket is a fractal object.

[desiresjab]

5 and 17 are a strange pair. They are both 4n+1. Phi/4 is 16, so there are plenty of factors of two left. Yet when the diagonal divides them it partitions them to 9 and 7. Since they are mutually rejective with that many factors of two left, that is all the diagonal could do.

Sometimes the diagonal is forced to parttion an odd number into two odd ones, as in 7 and 11, when there are only two factors of two (the minimum), or break an even one into two odds of different value, such as for 5 and 11, where there are only three factors of 2. But this is the first time I have seen a highly even number partitioned unevenly. The diagonal always gets it right. I have not had time to check the "natural" pairs for for 5 and 17 yet to see what they say. There are only 16 of them, so it will not be difficult.

[desiresjab]

5 and 17 are a strange pair. They are both 4n+1. Phi/4 is 16, so there are plenty of factors of two left. Yet when the diagonal divides them it partitions them to 9 and 7. Since they are mutually rejective with that many factors of two left, that is all the diagonal could do.

Sometimes the diagonal is forced to parttion an odd number into two odd ones, as in 7 and 11, when there are only two factors of two (the minimum), or break an even one into two odds of different value, such as for 5 and 11, where there are only three factors of 2. But this is the first time I have seen a highly even number partitioned unevenly. The diagonal always gets it right. I have not had time to check the "natural" pairs for for 5 and 17 yet to see what they say. There are only 16 of them, so it will not be difficult.

I did check those sixteen pairs of coordinates, and any conjecture crashes between the number of naturally expressible coordinates and the partitioning of lattice points. Those coordinates are merely the ones found strictly within WAXY. Only four coordinates out of sixteen are of this type for the two primes 5 and 17, and we know that number does not correspond to either value of the partition.

With certainty, we know that the diagonal for two 4n+3 primes will "thread," through an odd number of lattice points to partition them into one or other of WAY or YAX in unequal odd numbers.

With certainty, we know that a 4n+3 prime and a 4n+1 prime of lowest evenness for its kind (only divisible by 2 twice), will "thread" through an even number of points, partitioning them into odd halves which may be either equal or unequal, as far as we know.

With certainty, we know the two situations above were forced by a limited number of factors of 2.

In the Eisenstein rectangle lattice point graph for the two primes 5 and 17, the "lean" of the diagonal steals away three lattice points on the bottom row, but two of them get made up somewhere.

Whether or not the values of WAY and YAX can ever differ by more than two, becomes an interesting question in itself.

[YesNo]

You might try constructing proofs of the certain items if for no other reason than to get a foundation for future results. There is a theory of lattice points that I am unfamiliar with that might be a place to start.

I thought you found examples where the WAY and YAX differed by more than two lattice points. Perhaps not.

I'm working on a problem at the moment and trying to get Python to generate some examples or a solution. The claim is that the sum over n starting with 1 of (n-1)n is never prime. So the goal is to find a prime in that sequence of integers or find a covering congruence to show that the sequence has no primes.

I'll be happy if I can get a script to generate composites of this form up to n = 1000. I think I have a workable algorithm for that, but I don't even have a way to show that a covering set of primes actually covers all of the numbers in the sequence.

[desiresjab]

You might try constructing proofs of the certain items if for no other reason than to get a foundation for future results. There is a theory of lattice points that I am unfamiliar with that might be a place to start.

I thought you found examples where the WAY and YAX differed by more than two lattice points. Perhaps not.

I'm working on a problem at the moment and trying to get Python to generate some examples or a solution. The claim is that the sum over n starting with 1 of (n-1)n is never prime. So the goal is to find a prime in that sequence of integers or find a covering congruence to show that the sequence has no primes.

I'll be happy if I can get a script to generate composites of this form up to n = 1000. I think I have a workable algorithm for that, but I don't even have a way to show that a covering set of primes actually covers all of the numbers in the sequence.

I may have stated incorrectly once I had an example. I believe I have no examples of WAY and YAX with a difference of more than two lattice points. These problems get huge to generate by hand with relatively small primes. I have no mathematical software to assist.

Your problem sounds interesting, and either has an echo or a false echo of Fermat.

It is a snap to show that (p-1)(q-1)/4 is equivalent to Euler's totient function. Pretend that these two primes are really, really, really huge. We can always tell their types, but ascertaining whether one is a quadratic residue of the other may be next to impossible by hand. What can we do?

We also pretend we have a computer capable of multiplying (p-1)(q-1), in fact we will need one. Trusty division by four is our next step. Now we have something to look at. We can tell the species of this quotient, too.

If the quotient is already an odd number, we know the diagonal will produce an even and an odd number. We must have been dealing with two 4n+3 primes.

If the quotient is a 4n+3 number, at least the possibility if not the certainty of (1)(1) is preserved, though I see no way yet to determine if it is that or (-1)(-1) for very huge numbers.

[desiresjab]

Interestingly enough, suppose one intended a proof of quadratic reciprocity derived from the totient function. This would only work for odd primes. We have to use the formula (p-1)(q-1), where one or the other of the expressions is (2-1). A diagonal of the Eisenstein diagram of these dimensions will not make numerical sense.

The totient function is only a shortcut for odd primes, but what a shortcut it is.

[YesNo]

I may have stated incorrectly once I had an example. I believe I have no examples of WAY and YAX with a difference of more than two lattice points. These problems get huge to generate by hand with relatively small primes. I have no mathematical software to assist.

If you can use a spreadsheet you can check some of these for low numbers. I use Google sheets since it is convenient, in a cloud storage and free. Here's a link to a Google sheet I made some time ago about your conjecture: https://docs.google.com/spreadsheets/d/1RDz33GVXuKMledUjhHCRT67cmZ4QuKtPj9IRVKe3u10/edit?usp=sharing

It looks like 5 and 23 have a difference of 4. (Of course I might have constructed the sheet incorrectly.)

There are four tabs on the spreadsheet. On the Configuration tab there are entries for "Prime A" and "Prime B". Those are the only values to change. On the Lattice Points tab is a graph of how the lattice points look. The fractions represent deviations from 0 or the diagonal. The Twin Primes tab is a list of tests for twin primes. It is not fully filled out. The References tab are places I looked for lattice point information. The sheet is limited to numbers under 100.

You should be able to copy the spreadsheet from the link and modify that copy should you want to use something like this. You may need a Google account to set up Google drive if you don't have this already.



Your problem sounds interesting, and either has an echo or a false echo of Fermat.

I am planning on using Fermat's Little Theorem to simplify the calculations in Python. Basically I would be using ap-1 = 1 (mod p) for prime p. I have this set up on a Google sheet, but these easily get past the max size of integers on a spreadsheet. So I have to use something like Python. That's a free tool you also might find useful. I am still feeling my way around it, but I have programmed in many different languages for decades.

[desiresjab]

I did it twice by hand, trying to make sure. My rough graphs by hand are not ultimate arbiters, but it appears you may be right. Yet several lattice points are really hard to judge by eye. If we judge there to be 9 and 13 points respectively, and are wrong on one lattice point, that the brings the totals to a respectable 10 and 12, except we know that is wrong--neither one of these is the quadratic residue of the other, by the application of easy properties.

However, if we judge there to be 8 and 14 points respectively, being off by one point transfer would make both values negative again, and fulfill that needed condition.

Since you are using software, I tend to go with your results as more definitive in the judging-by-eye department.

[YesNo]

I tried it by hand initially, but then I realized I was making too many mistakes. I think the spreadsheet is correct, but I'm not sure. One of the problems with software is that it has many more pieces to check than a proof.

[desiresjab]

I tried it by hand initially, but then I realized I was making too many mistakes. I think the spreadsheet is correct, but I'm not sure. One of the problems with software is that it has many more pieces to check than a proof.

That is why we are still down in the engine room, looking for the fundamental mechanical principle of mere numbers everything relies on and quadratic reciprocity expresses. Sometimes investigations turn out to be superflous to solving the problem, but add to one's knowledge base. I would say that is the rule rather than the exception. We know now that the principle is not related to the number of lattice points expressible in WAXY through primitive combinatorial multiplication to generate coordinates. That was an important thing to get out of the way, once the idea came up.

Have the masters really captured everything there is know of QR from the ground level view? I cannot allay the suspicion that the mechaical principle is visible through all the gears, wires and steaming valves, if one stands exactly in the right place and bends down just so with a crane of the neck and peers through the complexity at the cause of it all.

Time was I was sure I had it, and up to a point I did. Once the combined totals of factors of 2 in (p-1) and (q-1) reach 24, however, the outcome of the diagonal split of this even number of lattice points in WAXY, cannot be predicted, though one has other rules, properties and laws to consult to usually clear up what the mutual reciprocity is, which is what one is usualy after.

We were looking at the number of points in WAXY and their coordinate names as a side issue. I no longer know if it is relevant.

The answer I am looking for, and the way I am looking for it, make a fantastically difficult yet solvable problem, I suspect. This is what I wanted. I see no need to move on. I keep learning more. The engine room is fine.

[YesNo]

I've started using pari/gp for calculations needing multi-precision arithmetic. It is also an algebra package primarily for number theory. It allows you to work with matrices and you could probably implement the WAXY pattern for integers larger than what the Google sheet allowed.

The gp part is a calculator and the pari part is a C library. I am more interested in the calculator. There are free C compilers available which could use the libpari library, but I'll give Python a chance first although I suspect pari might be faster. Here is the location of pari/gp: http://pari.math.u-bordeaux.fr/download.html

[YesNo]

Since Moffat's gravity theory made a prediction about gravity waves from the big bang which is different from what the Newton-Einstein theory would predict, I was looking at LIGO which showed the existence of gravitational waves last year. I am not sure what Moffat's prediction is, but at https://losc.ligo.org/about/ there is a tutorial about LIGO's recent findings with an interactive Jupyter notebook allowing you to play around with the data.

There are other alternative gravity theories. Moffat discusses them in "Reinventing Gravity". One of the benefits of a modified gravity is that dark matter would not be necessary and black hole singularities could be eliminated.

The inability to find dark matter and the observations of the rotational speed of galaxies are evidence that the Newton-Einstein theory of gravity is incorrect and needs modification. The observed movements of galaxies falsifies the Einstein gravitation theory without dark matter.

But all of these are theories or models of the universe. Even Moffat's theory is only a map. It is not the reality. Sometimes it is hard to keep the map and reality separate since the only way we can make sense out of reality is through a map.

[YesNo]

When Moffat discussed other alternative graitational theories and the problems in them, he wrote this regarding "quantum gravity", an attempt to combine gravitational theory with quantum theory: (John W. Moffat, "Reinventing Gravity", Harper Collins, 2008, page 142)


Some theorists simply claim that since gravity is observed and quantum mechanical effects are also observed that qualifies as enough experimental evidence that a quantum gravity theory is necessary.

The implication is that quantum gravity is not necessary. The problem with getting quantum gravity to work is to take any theory that works on the quantum level and getting predictable results that match what is observed about gravitation on the cosmic level.

If one doesn't need quantum gravity then there is no need for a "graviton", a quantum particle of force associated with gravitation like the photon is associated with electromagnetic radiation which so far has not been detected.

[desiresjab]

What results if any to report of Moffat's attempt at a time machine? I know he proposes a coil of lasers to produce warpage, where only light is involved in this bending instead of massive objects. He expects to find particles (marked somehow, I assume) when he performs his early experiments that he has already sent back in time to himself, which is mind bending. A single particle is what he is trying to send back or ahead, for now.

[desiresjab]

Whoops! Excuse me, please. I got names mixed up. The guy I was thinking of is Ron Mallett.

[YesNo]

I remember reading something by Ron Mallett regarding time travel some years ago, but I don't think time travel is possible. That would violate the second law of thermodynamics where we can only go from low entropy to high entropy, from past to future. By the way, I haven't run into any time travelers.

John Moffat has an interesting thing about time at t = 0 ("big bang") which he does not consider to be a singularity. He assumes there are two universes one going into the past and the other into the future. We don't know which one we are in. It helps avoid the singularity at t = 0. Other singularities such as black holes are also eliminated. And he doesn't need dark matter or a multiverse in which the anthropic principle can get us to where we are now. That is, it makes predictions which could be falsified or verified if LISA becomes operational and we can view gravitational waves from the origin of the universe.

But it is only a model. Its main goal is to fit the observations of acceleration in the galaxies that cannot be explained by Einstein's general relativity without assuming there is more matter in the universe than we can observe. It is useful or not if it can make accurate predictions.

[desiresjab]

By narrowing our look at Quadratic Reciprocity to twin primes only, we are able to initially highlight those two instances that interest us most, that is, where the larger twin is a quadratic residue of the smaller (and therefore the smaller is a residue of the larger, as well). We want to know what to expect of a set of twins at a mere glance.

Only 8n+1 and 8n+7 primes have 2 (the difference of any two twins) as a quadratic residue. In the case of 19 and 17, 19 is an 8n+3 prime which is a quadratic residue of 17, the 8n+1 prime. In the case of 73 and 71, 73 is a 9n+1 prime and 71 is an 8n+7 primes, whose difference is 2. Those should be the only two types of cases where the larger twin is a quadratic residue of the smaller.

Quadratic residues of 17 are 1, 4, 9, 16, 8, 2, 15, 13. Another way of expressing this group is: -9, -4, -2, -1, 1, 2, 4, 9,

The latter expression illustrates how the quadratic residues are grouped symmetrically around zero. Of course, we could always substitute 19 in for 2, to make it even more clear that 19 is a quadratic residue of 17. We can easily see, that is, when 6x6 is divided by 17, it leaves a remainder of 2, making 2, and thereby 19, a quadratic residue of 17.

Now let's make a list of the quadratic residues of 19.

1, 4, 9, 16, 6, 17, 11, 7, 5, It also looks like this:

-14 -13 -8, -3, -2, 1, 4, 9, 16.

The groupings are asymmetrical, for 19 is an 8n+3 prime, and of course therefore a 4n+3 prime.

We see that 17 is a quadratic residue of 19, as well. What we also see is that in either of these cases (an 8n+3 and an 8n+1, or a 9n+1 and an 8n+7), the two primes will be quadratice residues of each other. We further can note that in the case of the Legendre symbols for these two numbers, they will always be positive, so one is always multiplying two positive Legendre symbols together.

Of course the above cannot be the case in general for twin primes, but only for the two cases we looked at.

What happens for other twin prime combinations? Well, what has to happen? First of all, we think we can guarantee that all other combinations of twins will generate two negative Legendre symbols to multiply together to acheive positive 1. All we have to do is try a few.

11 and 13 are 8n+3 and 8n+5. Quadratic residues of 11 are:

1, 4, 9, 5, 3. An identical expression is: -8, -6, 1, 4, 9.

Quadratic residues of 13 are:

1, 4, 9, 3, 12, 10. An identical expression is: -10, -3, -1, 1, 4, 9.

Notice that neither group contains its twin in its residue set. Both Legendres will be negative, producing a positive upon multiplication.

The remaining case is 8n+5 with 8n+7. The twins 29 and 31 will fit this bill.

The residues of 29 are:

1, 4, 9, 16, 25, 7, 20, 6, 23, 13, 5, 28, 24, 22.

The residues of 31 are:

1, 4, 9, 16, 25, 5, 18, 2, 19, 7, 28, 20, 14, 10, 8.

Notice that once again, neither is in the other's quadratic residue set. The Legendres will be negative by themselves, producing a positive product.

The only case we did not explore was 9n+1 and 8n+7. Unfortunately the the smallest pair of twins with this form are 71 and 73, for which I do not care to calculate the sets. But I can guarantee they are quadratic residues of one another.

It appears that if primes, and in particular twin primes, are equally distributed by type, then half the pairs will be residues of one another and half will not be. In any case, for twin primes the combined Legendre symbols will always create positive reciprocity, whether it attains it through (1)(1) or (-1)(-1).

I conjecture for the moment that the latter [(-1)(-1)] occurs when only three factors of 2 are involved between the minus ones of the two twins.

This conjecture feels hopeful. If it were true, it would enable us to immediately "see" the characters of the separate Legendres involved. Everything seems pinned on the 8n+5 number, since it plays a part in both cases. If its minus one contains more than two factors of 2, I am saying it will never cut Eisenstein's lower triangle into odd halves.

Perhaps this is obvious, but I have to find a way to prove it or demonsrtate its truth or falsity clearly. The method would be to find any 8n+5 prime involved in a twinship, whose minus1 has more than two factors of two yet still divides Eisenstein's triangle into odd halves. For instance, 39+41 instead of 40+40, for eighty lattice points et al. This seems like an interesting question to pursue. We need to make a list of 8n+5 primes to see if any are both super even (more than two factors of 2) and involved in a twinship. It does not matter if our 8n+5 is the larger or the smaller of the twins.

5, 13, 29, 37, 53, 61.....

(101, 103)

It turns out that most of these primes are involved in a twinship. This is no way to proceed.

Wait. I have seen it. When 4 is added to any 8n number, it reduces its evenness by one factor of 2, the case with all 8n+5-1 numbers. Thank God for Gauss. This is exactly the condition we need for the conjecture to be true, and we see that the conjecture is indeed true.

A diagonal through WAXY in Eisensteins' rectangle in Wikipedia will always cut WAXY into two equal but odd numbers of lattice points whenever the diagram is for twin primes either of which is an 8n+5 number. The exact principles used here apply anytime one looks at any pair of primes of opposite type. This concludes the investigation.

* * * * *

It is now clear that 8n+5 numbers will always have two and only two factors of 2. This becomes clear when we tie these numbers to the ruler sequence, which expresses the degree of evenness of each consecutive even number, in other words, the number of factors of 2 it contains.

1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4...

Now I understand what Gauss was doing when he proceeded on from 4n numbers to investigate 8n numbers, and why.

At this time I am prepared to guarantee how a diagonal will apportion the lattice points of WAXY when the diagram is for twin primes, according to the types of primes involved. I am not quite prepared to guarantee how the diagonal will apportion lattice points for any two primes at all, based simply on their ruler function positions. But Halelujah anyway! I have learned much this time.

What I need to know now is this: When he have two super even (more than two factors of 2 apiece) 4n+1-1 numbers, I believe they will not always apportion the lattice points into two even sets just because they are super even. Only Eisenstein diagrams for rectangles with very low eccentricity (like twin primes) can guarantee what the cut will be. Eccentric rectangles may not always produce the result of two equal sets of lattice points. Which has me wondering if hugely eccentric rectangles can ever produce equal sets. Somehow, I suppose they can. But I really have no clue whether they can ever produce two unequal sets with an odd number of elements.

It would sure be nice if the ruler function ruled the whole law. For all I know, it does. I certainly hope it does. What that would mean is this: We could determine the cut, and thereby the Legendre symbols for any two primes at a glance. Let us pray the ruler function rules, which I doubt.

[desiresjab]

A look back into my own papers reveals the answer quickly. Both (5, 13) and (5, 17) arrive at their positive 1 through a multiplication of (-1)(-1), yet their minus ones all are highly even and 16 is super even, suggesting that past a certain level eccentricity plays a greater part than evennes in deciding the cut. This is now clear. As p/q varies from 1/1, a square, the distance of p/q from 1 is its eccentricity. I believe this ratio of p/q is the key factor in bringing the rest of the law under the reign of understanding. I can imagine a graph, two graphs intersecting, one for evenness, one for eccentricity. They have to cross somewhere. That is where one takes over dominance from the other.

[desiresjab]

To put it in an even smaller nutshell: The minus ones of the four types of 8n+z primes all have their own evenness which is perfectly predictable. Only the minus ones of 8n+1 type primes are ever higher than two factors of 2. Just as the ruler function shows, all the action that changes is in the 8n slots, making 8n+1 primes the only primes whose minus one can be super loaded with factors of 2. How super evenness and eccentricity of the Eisenstein rectangle interact, is now the question.

[desiresjab]

I will now make a conjecture which I myself may be able to quickly dash.

Any two 4n+1 primes far enough out the number line, one of which is an 8n+1 prime, will make the Eisenstein cut into two identical and even sets. Even separated by four instead of two, if they are far enough out the number line the eccentricity of their rectangle will approach zero, and should force the apportioning of the diagonal into two even and equal sets. It only takes one counter example to dash the conjecture to smithereens.

Must two highly even primes far out the number line really always be quadratic residues of one another merely because they are relatively close together? Hmmm. This sounds very suspicious. But we shall see.

[desiresjab]

Confirmed already! How? Because it was trivial after all. If one 4n+1 prime overlaps another by 4, four is a square, so they will obviously both have to be residues of the other, since one of them is at a glance known to be.

[desiresjab]

By the way, a previous post did not post, and the previous conjecture was easily solved, since the two primes in question overlapped by 4, a square number.

Which leads us to the next question, actually observation. Yeah, it is like a took a math pill tonight.

Suppose we have two super even 8n+1 primes. One is relatively close to zero, such as 17, for instance; the other is tremendously far out the number line. As long as the distance between them, their difference, the overlap number, is an obvious square number that we can see, we are guaranteed positive reciprocity positively gained, that is, by multiplying (1)(1) instead of (-1)(-1).

So in the case of this hugely eccentric rectangle we are guaranteed positive reciprocity positively gained. But do we know if the quadrant rectangle WAXY in Eisenstein's diagram on Wikipeja will be divided by the diagonal into two equal sets? Unfortunately, we do not. We only can guarantee that both sets will contain a positive number of lattice points, not that they will be equal. We cannot even say the difference between lattice points of the two sets cannot exceed two. We believe this is the case, but cannot prove or demonstrate it to our own satisfaction yet. We is me, apparently.

[YesNo]

I am glad to see you back doing number theory. I am off and on thinking of the Sierpinski sequences and whether I can form coverings of them. I have been thinking about using Python to generate a cover, but I keep getting distracted.

[desiresjab]

I am glad to see you back doing number theory. I am off and on thinking of the Sierpinski sequences and whether I can form coverings of them. I have been thinking about using Python to generate a cover, but I keep getting distracted.

That gasquet is often used to illustrate similarity across scale in books on fractal geometry. I have seen it, and about all I know is that it has fractal properties.

It is good to be back on QR, especially since I am making progress. My current attempt is to find something in the behavior of highly even numbers that distinguishes them. Only 8n type numbers are super even (more than two factors of 2).

I am beginning to suspect there might be no defining behavior that sets them apart other than what I have already stated about the role of "degree of evenness", otherwise I would already have found it in the literature. This likely why 4n types are the only ones used in the formal definitions.

Something I consider quite important that I learned last night from graphing is that the difference between lattice point sets can exceed 2. For the primes (5, 41), one set has 16 points and the other 24. Now we know any number can be this difference, depending only on the eccentricity of the rectangle. For me this is a huge breakthrough.

The thing about 8n numbers being the only ones with super evenness I should have realized long ago. I have been in possession of the ruler function for about a year and only just now have put the thoughts together.

[desiresjab]

The question might be asked, "Why explore this in a cosmology thread?" Cosmology is what I think it is. I am trying to look into the mind of the creator, to quote an idea found in Peter Martinson's paper on QR. Every time one understands something about math they previously did not, it amounts to looking into the creator's mind and methods. The deeper the proposition, the deeper one must look into the creator to understand it