Cosmology

[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...ciprocity.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)=3^1/2(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.