Cosmology

[desiresjab]

I didn't know about these markings, but it supports the idea of neuroplasticity which is part of a recent kind of evidence of how the mind affects the brain rather than the other way around.



Here is a link to a variety of proofs: https://en.wikipedia.org/wiki/Proofs...little_theorem

I am familiar with the ones for modular arithmetic and the proof using the binomial theorem. I was unaware of Golomb's combinatorial proof: http://www.cimat.mx/~mmoreno/teachin...Little_Thm.pdf. One thing Golomb asks which is important for these proofs to make sure they are correct is where do they use the hypothesis that p is prime since the result is not in general true for all integers.

I am still trying to understand your proof about tiling as well as the dynamical system proof mentioned in the link of proofs above.

I have looked at all these proofs before, in keeping with my many perspectives philosophy. The combinatorial proof is not bad, but the simplest proof is the first proof given by way of modular arithmetic. All you have to know beforehand is that when you multiply the elements of the set {1, 2, 3, 4,....n} by a constant a, the original set is merely reproduced in a different order by the multiplication. This is where you get to Wilson's theorem from, as well.

[YesNo]

I have looked at all these proofs before, in keeping with my many perspectives philosophy. The combinatorial proof is not bad, but the simplest proof is the first proof given by way of modular arithmetic. All you have to know beforehand is that when you multiply the elements of the set {1, 2, 3, 4,....n} by a constant a, the original set is merely reproduced in a different order by the multiplication. This is where you get to Wilson's theorem from, as well.

That different order is what leads me to have doubts about the proof although I know the result is correct. It does use the hypothesis that p is prime.

Fortunately, these thoughts are interesting in themselves, for I don't see how they connect with what I said about Godly constraints. No matter.

What I thought was similar was that both of you use mathematics more than I would. I would like to see something more conscious involved.

[desiresjab]

That different order is what leads me to have doubts about the proof although I know the result is correct. It does use the hypothesis that p is prime.

I lost a long reply because my computer froze up. Just as well. It was probably too pedantic and meandering.

The question is whether the set {0, 1, 2, 3, 4, 5, 6,...n} will actually reproduce itself in its entirety when each member is multiplied by the same constant. Remember that under a modulus numbers have nowhere to go when multiplied, except to one or another residue class of the reisdue system. They are trapped, they do nothing but cycle.

For modulus 7, 94827165103984648126484356, is in one or another of the 7 seven residue classes. That much is gauranteed, because all integers are. That huge number above can always be reduced to one of the reside classes and its most basic, i.e. smallest representative of the class.

{49, 1, 2, 38, 4, 705, 6}, is also a complete residue system of 7, because each class is represented once, even though some of the representatives are not fully reduced. That makes no difference. All numbers in a residue class are exactly equivalent, and may be substituted for one another at any point in calculations.

Below is the real key, the short expo.

If x and y are already congruent, then ax and ay will still be congruent, i.e. belong to the same residue class as each other after the multiplication, though it may now be a different class they are in together. This is one of the fundamental properties of conguences.

The property works in reverse, as well. The members of the set were all mutually incongruent to each other to begin with, because they belonged to different residue classes. Multiplied by the same constant a, they must all remain incongruent, as each cycles around the clock face according to the multiplier, to its eventaul slot.

Since they must all remain mutually incongruent after the multiplication, they are trapped again, the seven different products have no choice but to represent each residue class, lest two of them be congruent, which members of different residue classes cannot be, by definition.

I enjoyed that. I really had to think it through. I am lucky my computer crashed three times, as it turns out.

Everything that needs to be understood with regards to "a different order" is contained in the last three paragraphs of my last post.

[YesNo]

I lost a long reply because my computer froze up. Just as well. It was probably too pedantic and meandering.

The question is whether the set {0, 1, 2, 3, 4, 5, 6,...n} will actually reproduce itself in its entirety when each member is multiplied by the same constant. Remember that under a modulus numbers have nowhere to go when multiplied, except to one or another residue class of the reisdue system. They are trapped, they do nothing but cycle.

For modulus 7, 94827165103984648126484356, is in one or another of the 7 seven residue classes. That much is gauranteed, because all integers are. That huge number above can always be reduced to one of the reside classes and its most basic, i.e. smallest representative of the class.

{49, 2, 38, 4, 505, 6}, is also a complete residue system of 7, because each class is represented once, even though some of the representatives are not fully reduced. That makes no difference. All numbers in a residue class are exactly equivalent, and may be substituted for one another at any point in calculations.

Shouldn't there be 7 elements in the set making the elements congruent to {0,1,2,3,4,5,6}?

Below is the real key, the short expo.

If x and y are already congruent, then ax and ay will still be congruent, i.e. belong to the same residue class as each other after the multiplication, though it may now be a different class they are in together. This is one of the fundamental properties of conguences.

That makes sense because given a prime p and x = y mod p then given an integer a, ax = ay mod p. In this case a could equal 0.

The property works in reverse, as well. The members of the set were all mutually incongruent to each other to begin with, because they belonged to different residue classes. Multiplied by the same constant a, they must all remain incongruent, as each cycles around the clock face according to the multiplier, to its eventaul slot.

Going in the other direction if x is not congruent to y mod a prime p then multiplying x and y by a = 0 would make them congruent. The proof in the link avoids a = 0 for a^p-1 = 1 mod p by making sure 0 < a < p. However a^p = a mod p works for a = 0 since one can factor out the a.

So my question would be given any prime p, how do we know there aren't other residues that act like 0 in the set of residues mod p besides 0?

Now I know there is only one element that acts as a zero as well as only one element that acts as a unit (1), but I wonder if this requires some sort of proof or can it be assumed at this point?

Edit: This does seem to be where we need the hypothesis that p is a prime. If p were 4, then 2*2 = 0 mod 4. These residue classes mod a prime form finite fields: https://en.wikipedia.org/wiki/Finite_field

Since they must all remain mutually incongruent after the multiplication, they are trapped again, the seven different products have no choice but to represent each residue class, lest two of them be congruent, which members of different residue classes cannot be, by definition.

[desiresjab]

You are right. I did miss an element in a set I listed. Sorry about that. I went back and corrected it.

The zero class of residues is generally left out of many procedures in the business, I believe, because it has no inverse and some other reasons. I did not leave the zero class out, though, because 49≡0 (mod 7). I left out one of the other classes by oversight.

I may have made other mistakes.

The fundamental property that if a≡b (mod p), then ax≡bx (mod p), has an analogue with addition, for it is also fundamental that if a≡b (mod p), then a+x≡b+x (mod p).

It is also true with exponents. If a≡b (mod p), then aⁿ≡bⁿ (mod p)

Being able to accept with clarity just these three properties, is essential. They are powerful tools and lead many places.

I have satisfied myself with respect to Fermat's little theorem. I feel I have seen to the bottom of the well on that one. It has been reduced to a compact visualization.

Seeing to the bottom of the well on quadratic reciprocity may not be possible for me. The kind of visualization I seek may not be a realistic possibility. I think such understanding may involve seeing to the bottom of the well on Eisenstein's geometric proof, for he has probably already reduced it to its simplest representation. The innocent eye will not even detect a relation between Eisenstein's lattice points in a rectangular array and the law of QR, that is how far he has gone. To understand his proof, an understanding of other proofs and a familiarity with their notations is essential. I am pretty sure you have to have this intimate familiarity to see to the bottom of the well, even with Eisenstein's deceptively simple proof. When I can do that, or have devised my own representation, only then may I be able to say I have seen to the bottom of the well with respect to QR. That would be a nice feeling to experience, and I wonder if I am going to have it.

Think how powerful the mind of Artin had to be, to finally conquer general reciprocity, when even little ol' quadratic reciprocity is so tangled and tough. How monumental was that task?

[YesNo]

I have satisfied myself with respect to Fermat's little theorem. I feel I have seen to the bottom of the well on that one. It has been reduced to a compact visualization.

Seeing to the bottom of the well on quadratic reciprocity may not be possible for me. The kind of visualization I seek may not be a realistic possibility. I think such understanding may involve seeing to the bottom of the well on Eisenstein's geometric proof, for he has probably already reduced it to its simplest representation. The innocent eye will not even detect a relation between Eisenstein's lattice points in a rectangular array and the law of QR, that is how far he has gone. To understand his proof, an understanding of other proofs and a familiarity with their notations is essential. I am pretty sure you have to have this intimate familiarity to see to the bottom of the well, even with Eisenstein's deceptively simple proof. When I can do that, or have devised my own representation, only then may I be able to say I have seen to the bottom of the well with respect to QR. That would be a nice feeling to experience, and I wonder if I am going to have it.

Don't give up hope. However, if you wanted to see to the bottom of the well of Joyce's "Finnegans Wake", I would recommend despair. The bottom may be a lot shallower than quadratic reciprocity.

I checked this on quadratic reciprocity: https://en.wikipedia.org/wiki/Quadratic_reciprocity

It looks like it starts with Fermat's little theorem relating a^p-1 = 1 mod p and then asks what one can say about a^(p-1)/2 = +/- 1 mod p. I don't understand it. Nor do I understand Artin's generalization, but perhaps we can try to clarify that for each other.

[desiresjab]

Don't give up hope. However, if you wanted to see to the bottom of the well of Joyce's "Finnegans Wake", I would recommend despair. The bottom may be a lot shallower than quadratic reciprocity.

I checked this on quadratic reciprocity: https://en.wikipedia.org/wiki/Quadratic_reciprocity

It looks like it starts with Fermat's little theorem relating a^p-1 = 1 mod p and then asks what one can say about a^(p-1)/2 = +/- 1 mod p. I don't understand it. Nor do I understand Artin's generalization, but perhaps we can try to clarify that for each other.

The theory of primitive roots is another interesting study in number theory. Fermat's little theorem does not say whether a^(p-1) is the first power that equals one under the modulus. There could have been earlier powers that are equal to 1. Primitive roots are equal to 1 for the first time at (p-1). So primitve roots are special and a whole theory is built around them.

I cannot even see beneath the water in Finnegan's Wake. Two or three hundred years from now people will still feel the same about that book. Every human being on earth with a clear understanding of Relativty would be more likely than everyone with an understanding of Finnegan's Wake. Now that book is opaque.

The first to fully understand that book will probably be a meatie with integrated implants.

You are right about the multiplication by zero. My own proof relied on a factorization which allowed me to show that some of the factors belonged to the zero class of residues (remainders).

General reciprocity means all of them, the cubic, the quatric, the quintric... What are the laws of general reciprocity, not just quadratic, you see? Artin manged to untangle that. He is one of the great math men of all time that you never hear about. General reciprocity was a problem from Hilbert's original famous list. Finding the solution of any one of those problems guarantees one immortality. Many of those problems have now been solved. In one article Artin is referred to as the preemminent algebraist of the 20^th century.

There are many angles to view QR from. They are all correct but they all illustrate different aspects of it. It has many different equivalent statements.

Basically, it compares two primes to find out if either is in the other's quadratic residue set. I know that is a mouthful. Let us compare 3 and 5, for the sake of simplicity. Are there any numbers in the baisc residue set of 3 which when squared are equal to 5 (mod 3)? But 5 is equal to 2 (mod 3). Are there any numbers under three which equal 2 when squared, then? There are only 1 squared and 2 squared, which both equal 1 (mod 3).

Now we ask the reverse question--can the number 3 be found when the numbers less than 5 are squared (mod 5)? Let us look. 1²=1, 2²=4, 3²=4, 4²=1 all mod (5).

These numbers (3 and 5) are not quadratic residues of each other, since neither can be found in the other's quadratic residue set. This gives them, when multipled together, as in the Legendre symbol, a value of 1, because (-1)(-1)=1.

Another way of stating the general law is that if either of the primes being compared is a 4n+1 type prime, then both primes are either in the other's set, or both are not.

In one species of case, where we have two primes of 4n+3 variety, one will be in the other's quadratic residue set, and the other will not be in the other's. In this case we have a kind of quadratic irreciprocty, as I like to call it, and the value of the Legendre will be -1, since (-1)(1)=-1. Only in the case of two 4n+3 primes will the Legendre symbol ever equal -1.

two 4n+3 types=-1
one of each type=1
two 4n+1 types=1

When we compared 3 and 5, we had one of each type. We only need to find one value, in this case, because the other is guaranteed to have the same "character" as its companion when the two primes are of different types. One calculation is always easier than the opposite way. The easy calculation always implies the answer to the other prime.

The same reasoning applies when comparing two 4n+3 type primes. Do the easy calculation, and the other value is automatically known to be of opposite "character" to that one.

There is a strong concept of periods involved in reciprocity, which I do not have a full grip on yet. Once, I thought I had it rassled down and pinned, but my hold was illegal. Modulus rings are all about periods. They have torsion, which means multiplying by a larger number can make them smaller sometimes. Normal arithmetic is not entirely applicable in modulus rings, obviously. A firmer understanding of which periods affect reciprocity and how, would clean things up a bit for myself, methinks. Actually, QR has me tired for the moment, but I will cycle back in a few days refreshed. Repeated seiges must win a war of attrition.

The fact that prolonged seiges are necessary, means I am dumb, neither a first class nor a second class mathematician, when one considers, my God, that Gauss gave as criterion for a first class mathematician an immediate understanding of Euler's formula e^iπ+1=0, where that exponent that comes out looking like a weird M is actually the Greek letter pi, and i is the imaginary number the square root of -1. Also, e is the constant found universally in nature. Mathematically, e is a function which is its own derivative and integral, which makes it really cool, and it is also a transcendental number!

There is more to the complexity of QR, though the 4n+1 and 4n+3 rules stand fast through all.

[YesNo]

The theory of primitive roots is another interesting study in number theory. Fermat's little theorem does not say whether a^(p-1) is the first power that equals one under the modulus. There could have been earlier powers that are equal to 1. Primitive roots are equal to 1 for the first time at (p-1). So primitve roots are special and a whole theory is built around them.

I noticed there is a conjecture Artin made about primitive roots: https://en.wikipedia.org/wiki/Artin%...rimitive_roots

Maybe we can try to prove it for a = 3. I hear it hasn't been shown even for one value.

I cannot even see beneath the water in Finnegan's Wake. Two or three hundred years from now people will still feel the same about that book. Every human being on earth with a clear understanding of Relativty would be more likely than everyone with an understanding of Finnegan's Wake. Now that book is opaque.

The first to fully understand that book will probably be a meatie with integrated implants.

Or someone with a computer with nothing better to do.

Here's a site with a lot of integer sequences: https://oeis.org/A005596

General reciprocity means all of them, the cubic, the quatric, the quintric... What are the laws of general reciprocity, not just quadratic, you see? Artin manged to untangle that. He is one of the great math men of all time that you never hear about. General reciprocity was a problem from Hilbert's original famous list. Finding the solution of any one of those problems guarantees one immortality. Many of those problems have now been solved. In one article Artin is referred to as the preemminent algebraist of the 20^th century.

There are many angles to view QR from. They are all correct but they all illustrate different aspects of it. It has many different equivalent statements.

Basically, it compares two primes to find out if either is in the other's quadratic residue set. I know that is a mouthful. Let us compare 3 and 5, for the sake of simplicity. Are there any numbers in the baisc residue set of 3 which when squared are equal to 5 (mod 3)? But 5 is equal to 2 (mod 3). Are there any numbers under three which equal 2 when squared, then? There are only 1 squared and 2 squared, which both equal 1 (mod 3).

Now we ask the reverse question--can the number 3 be found when the numbers less than 5 are squared (mod 5)? Let us look. 1²=1, 2²=4, 3²=4, 4²=1 all mod (5).

These numbers (3 and 5) are not quadratic residues of each other, since neither can be found in the other's quadratic residue set. This gives them, when multipled together, as in the Legendre symbol, a value of 1, because (-1)(-1)=1.

Another way of stating the general law is that if either of the primes being compared is a 4n+1 type prime, then both primes are either in the other's set, or both are not.

In one species of case, where we have two primes of 4n+3 variety, one will be in the other's quadratic residue set, and the other will not be in the other's. In this case we have a kind of quadratic irreciprocty, as I like to call it, and the value of the Legendre will be -1, since (-1)(1)=-1. Only in the case of two 4n+3 primes will the Legendre symbol ever equal -1.

two 4n+3 types=-1
one of each type=1
two 4n+1 types=1

When we compared 3 and 5, we had one of each type. We only need to find one value, in this case, because the other is guaranteed to have the same "character" as its companion when the two primes are of different types. One calculation is always easier than the opposite way. The easy calculation always implies the answer to the other prime.

The same reasoning applies when comparing two 4n+3 type primes. Do the easy calculation, and the other value is automatically known to be of opposite "character" to that one.

There is a strong concept of periods involved in reciprocity, which I do not have a full grip on yet. Once, I thought I had it rassled down and pinned, but my hold was illegal. Modulus rings are all about periods. They have torsion, which means multiplying by a larger number can make them smaller sometimes. Normal arithmetic is not entirely applicable in modulus rings, obviously. A firmer understanding of which periods affect reciprocity and how, would clean things up a bit for myself, methinks. Actually, QR has me tired for the moment, but I will cycle back in a few days refreshed. Repeated seiges must win a war of attrition.

The fact that prolonged seiges are necessary, means I am dumb, neither a first class nor a second class mathematician, when one considers, my God, that Gauss gave as criterion for a first class mathematician an immediate understanding of Euler's formula e^iπ+1=0, where that exponent that comes out looking like a weird M is actually the Greek letter pi, and i is the imaginary number the square root of -1. Also, e is the constant found universally in nature. Mathematically, e is a function which is its own derivative and integral, which makes it really cool, and it is also a transcendental number!

There is more to the complexity of QR, though the 4n+1 and 4n+3 rules stand fast through all.

So why do people care about these reciprocity relationships?

Regarding e^iπ+1=0 wouldn't this be just (-1,0) on the unit circle? With e^ix = cos x + i sin x? https://en.wikipedia.org/wiki/Euler%27s_formula

[desiresjab]

I noticed there is a conjecture Artin made about primitive roots: https://en.wikipedia.org/wiki/Artin%...rimitive_roots

Maybe we can try to prove it for a = 3. I hear it hasn't been shown even for one value.

So why do people care about these reciprocity relationships?

Regarding e^iπ+1=0 wouldn't this be just (-1,0) on the unit circle? With e^ix = cos x + i sin x? https://en.wikipedia.org/wiki/Euler%27s_formula

I have not read about Artin's conjecture yet. I do not think I want to try solving anything even for 3 that the greats have failed to answer. One such problem on a man's calendar is quite enough, and I already have such a problem. It is called Brocard's problem, and I have been working on it for years. Investigating it has caused me to study in detail the classical elements of number theory, so the project, though hopeless, has not been fruitless.

As for the Euler equation, that is what it represents all right. If you saw that immediately, and then exactly what the trig function means, you will be a first rate mathematician, son. Congratulations.

It is now evident I am a thorough amateur. By thorough I must mean strictly. But I am an assiduous one, perhaps too dumb to relent.

Now I need a new project all right. Seeing to the bottom of the well on something as complex as QR requires some projects in between, otherwise you are just stuck in one place and are not learning anything. My method has been to study those areas I think might yield fruit on my Brocard project. Since I can never reasonably hope to solve Brocard's problem, what I can get out of the pursuit is whatever I can pick up that might relate to understanding it better. This leads a man far afield into pleasurable pursuits of learning, and at least partially justifies the obsession with an unsolved problem which some of the ATG's of mathematics have looked at without success.

Next I will take a look at your Artin link. I am well travelled on the other site. Sequences and series is one of my favorite aspects of math. The historical importance of series cannot be over valued. Studying series is one of the most fun things a human can do, at least this silly human finds it exhilerating.

Now I must hunt for something which I do not understand and which looks aesthetically appealing and relevant. In between, I write poetry, stories and novels, just like the other folks on the forum do. I seldom try to get anything published because I am too busy sorting everything at once. Dang it! Math is my hobby. A slow individual has to think a long time on these difficult matters to get them even semi-sorted out. The payoff is in ecstasy, though, man. I don't know why that is. I write better than I cipher, but ciphering will just not go away. I like performing certain acts, such as writing novels and going to the well with equations. Promoting them is boring as hell.

You could write something you knew was world class and have no success at all convincing editors and publishers of this. But if one ever did crack an unsolved problem, no editor or publisher could deny the acheivement with the flick of his wrist toward the waste basket.

That is one great difference: Initially cracking through to the world of literature depends soley upon the opinions of a few important people, whereas cracking the world of math depends soley upon fact which others may not even dispute. In math you cannot be shut out. Even if you are killed tomorrow like Galois, your acheivement lives on as long as your proof was written down. How many great pieces of literature were thrown irretrievably into the dust bin of time, ignored and lost? More than a few, I personally suspect.

[desiresjab]

I looked at the Artin conjecture. A couple of things to notice:

2. Under the conditions that a is not a perfect power and that a0 is not congruent to 1 modulo 4, this density is independent of a and equals Artin's constant which can be expressed as an infinite product...

Is the same thing as saying that a must be a 4n+3 type prime (with the apparent exception of 2, of course), a subject we just discussed. It is surprisng how much this idea crops up in high powered research. This may be an instance of the wide ranging influence of QR in other areas. You just asked why mathematicans were so concerned with QR. QR is centrally placed in number theory, just like the Pythagorean theorem is to normal algebra, geometry and trig--really important! It touches almost everything, but its hand is often concealed.

The other thing to notice in the article is the importance again of the Riemann hypothesis to eventual solutions. The Riemann conjecture has to be by far the most important unsolved problem in all of mathematics. If it fell, many famous unsolved problems would fall right behind it, for all they need to be complete is that the Riemann hypotheis be true.

Fermat's last theorem was the most famous mathematics problem ever, probably, and solving it justifiably grants Wiles immortality, but a host of other important solutions did not fall right behind it, as they will when the Riemann hypothesis is proven.

[desiresjab]

I don't want to hog the airwaves here, but I wanted to try and answer Yes/No's question as to the importance of QR. Besides the generalized answer that it is centrally located in number theory, I should state that it relates directly to the solution of quadratic equations in the algebraic structure of the ring of integers under a modulus.

When, why and why not, do quadratic equations have solutions in this algebraic structure, and what are those solutions?

Whether x²≡7 (mod 11), for instance, is souluble depends on QR in this algebra. This child of Gauss is as close as one can get to something like the quadratic formula of normal algebra.

Quadratic research is an ongoing thing. Many a Phd dissertation covers some aspect of it.

Quadratic equations are one of those mathematical objects we can clutch and cling to as sure things--we can get solutions. No wonder they are so important in the history of math and retain their importance. Galois told us two centuries ago there were no sure things beyond degree four in equations, no general method to extract solutions to equations of degree five and higher. We got what we got.

Whether God could have built a universe where general methods for equations of degree five and beyond are as plain and simple as quadratic issues--even that is an open question.

[YesNo]

I have not read about Artin's conjecture yet. I do not think I want to try solving anything even for 3 that the greats have failed to answer. One such problem on a man's calendar is quite enough, and I already have such a problem. It is called Brocard's problem, and I have been working on it for years. Investigating it has caused me to study in detail the classical elements of number theory, so the project, though hopeless, has not been fruitless.

It's not hopeless. Understanding Finnegans Wake is hopeless.

You might also try hypnosis.

I haven't heard of Brocard's problem: https://en.wikipedia.org/wiki/Brocard%27s_problem I assume the challenge is to find values of n such that n! + 1 is a perfect square. I see there are only 3 solutions. Is the goal to find a fourth or prove that they have all been found?

As for the Euler equation, that is what it represents all right. If you saw that immediately, and then exactly what the trig function means, you will be a first rate mathematician, son. Congratulations.

It is now evident I am a thorough amateur. By thorough I must mean strictly. But I am an assiduous one, perhaps too dumb to relent.

All I know is what I remember from first year calculus. What these things mean over the complex numbers is not something I can visualize.

Now I need a new project all right. Seeing to the bottom of the well on something as complex as QR requires some projects in between, otherwise you are just stuck in one place and are not learning anything. My method has been to study those areas I think might yield fruit on my Brocard project. Since I can never reasonably hope to solve Brocard's problem, what I can get out of the pursuit is whatever I can pick up that might relate to understanding it better. This leads a man far afield into pleasurable pursuits of learning, and at least partially justifies the obsession with an unsolved problem which some of the ATG's of mathematics have looked at without success.

Next I will take a look at your Artin link. I am well travelled on the other site. Sequences and series is one of my favorite aspects of math. The historical importance of series cannot be over valued. Studying series is one of the most fun things a human can do, at least this silly human finds it exhilerating.

Now I must hunt for something which I do not understand and which looks aesthetically appealing and relevant. In between, I write poetry, stories and novels, just like the other folks on the forum do. I seldom try to get anything published because I am too busy sorting everything at once. Dang it! Math is my hobby. A slow individual has to think a long time on these difficult matters to get them even semi-sorted out. The payoff is in ecstasy, though, man. I don't know why that is. I write better than I cipher, but ciphering will just not go away. I like performing certain acts, such as writing novels and going to the well with equations. Promoting them is boring as hell.

We are similar. You should try the poetry contests. You can always put what you post here in your blog or a book of poems later if you want.

You could write something you knew was world class and have no success at all convincing editors and publishers of this. But if one ever did crack an unsolved problem, no editor or publisher could deny the acheivement with the flick of his wrist toward the waste basket.

That is one great difference: Initially cracking through to the world of literature depends soley upon the opinions of a few important people, whereas cracking the world of math depends soley upon fact which others may not even dispute. In math you cannot be shut out. Even if you are killed tomorrow like Galois, your acheivement lives on as long as your proof was written down. How many great pieces of literature were thrown irretrievably into the dust bin of time, ignored and lost? More than a few, I personally suspect.

You still need someone to recognize that your proof was correct. Or someone else will have to re-create it.

[YesNo]

I looked at the Artin conjecture. A couple of things to notice:

2. Under the conditions that a is not a perfect power and that a0 is not congruent to 1 modulo 4, this density is independent of a and equals Artin's constant which can be expressed as an infinite product...

Is the same thing as saying that a must be a 4n+3 type prime (with the apparent exception of 2, of course), a subject we just discussed. It is surprisng how much this idea crops up in high powered research. This may be an instance of the wide ranging influence of QR in other areas. You just asked why mathematicans were so concerned with QR. QR is centrally placed in number theory, just like the Pythagorean theorem is to normal algebra, geometry and trig--really important! It touches almost everything, but its hand is often concealed.

The other thing to notice in the article is the importance again of the Riemann hypothesis to eventual solutions. The Riemann conjecture has to be by far the most important unsolved problem in all of mathematics. If it fell, many famous unsolved problems would fall right behind it, for all they need to be complete is that the Riemann hypotheis be true.

Fermat's last theorem was the most famous mathematics problem ever, probably, and solving it justifiably grants Wiles immortality, but a host of other important solutions did not fall right behind it, as they will when the Riemann hypothesis is proven.

It seems like the conjecture has been almost completely proved except for identifying the one or two exceptions that do not work. They would be either 3, 5, or 7.

It seems that -1 would not work since it could be the primitive root for only Z₃^x since it flips from -1 to 1 and back again giving only two distinct units. Also squares would not work since the most they could generate are half of the units and their square root would be the primitive root. So I can see why -1 and the squares are excluded. One already knows there can be only finitely many primes, if any, having them as a primitive root.

So, to proceed we would have to get the proofs by Roger Heath-Brown and R. Gupta and M. Ram Murty.

Then the challenge would be to actually construct the S(a) sets.

Edit: After looking at some of the other papers besides the Wikipedia one there might be more than a couple numbers which do not follow the conjecture. I think all that has been shown are that infinitely many numbers do, but which ones do not is not known. It is possible that the set of numbers that do not have infinitely many primes for which they are primitive roots is the set eliminated in Artin's hypothesis namely -1 or squares.

Of course, I might be totally confused about all of this. I am still putting the pieces of this jigsaw puzzle on the table.

[desiresjab]

It seems like the conjecture has been almost completely proved except for identifying the one or two exceptions that do not work. They would be either 3, 5, or 7.

It seems that -1 would not work since it could be the primitive root for only Z₃^x since it flips from -1 to 1 and back again giving only two distinct units. Also squares would not work since the most they could generate are half of the units and their square root would be the primitive root. So I can see why -1 and the squares are excluded. One already knows there can be only finitely many primes, if any, having them as a primitive root.

So, to proceed we would have to get the proofs by Roger Heath-Brown and R. Gupta and M. Ram Murty.

Then the challenge would be to actually construct the S(a) sets.

Edit: After looking at some of the other papers besides the Wikipedia one there might be more than a couple numbers which do not follow the conjecture. I think all that has been shown are that infinitely many numbers do, but which ones do not is not known. It is possible that the set of numbers that do not have infinitely many primes for which they are primitive roots is the set eliminated in Artin's hypothesis namely -1 or squares.

Of course, I might be totally confused about all of this. I am still putting the pieces of this jigsaw puzzle on the table.

Something to stay cognizant of is that (p-1) and -1 are the same thing, they represent the same class, therefore are identical. I can only delve deeply into problems that attract me strongly. Many problems are quite interesting. But a problem has to have a certain form, a certain look before I devote myself to it. For one thing, I would not look seriously at general reciprocity until I was thoroughly comfortable with QR. I am better off plugging through number theory textbooks, I believe, than taking on multiple unsolved problems. But all unsloved problems are of general interest to me.

And by the way, Merry Christmas to you and everyone on the forum. Simulations can be merry, can't we?

[YesNo]

Merry Christmas! Even if you aren't a simulation!