1. Originally Posted by Dreamwoven
All this advanced maths is way above my head, like a new language.
It is above my head, too, DW. That is why I am trying to understand it. A famous mathematician once said math is an unnatural way to think. True but strange, since it seems to be the language of natural things, at the very least their superficial language. For people like Euler and Gauss it is probably not unnatural. For world class mathematicians of any era, I have no idea whether it is a natural way. I suppose it must become so after a while. Even lowly I can notice a difference in my own abilities after having stuck with math assiduously for the last year.

Some of the misconceptions I had along the way and had to correct--why, it's preposterous and laughable. Gauss or Euler or the next level down from them would never have such problems on the same material, I imagine. They invented half the stuff I am still trying to understand well. All they had to guide them were their own instincts.

* * * * *

Anyone with a cosmological thought should voice them here. This is not a math thread but only seems so at the moment. Thoughts on the subject that are not math-oriented are not interrupting anything but adding to the discussion's breadth. I do not know peoples' ages on here. I know death begins to preoccupy the mind after a certain age. Death and cosmology go together.

* * * * *

Certain questions from months ago are still haunting me. Are types of universes possible which are logically impossible to us? I don't know how one would ever answer that.

2. I see now how e, pi and i got into those number theory equations. Like Yes/No said they are for finding the roots of unity for any xn-1=0, a long standing problem historically. I found an abstract algebra video where the instructor explained it with the unit circle for the roots of uinity of x5-1=0. He showed how to get roots of unity every 72 degrees, by multiples of one initial position equation.

They had me wondering, too. That is why I italicized them. Roots of unity are slightly different than the roots of equations we studied in high school, where we would surely be able to find two roots to x5-1=0, but what about the other three, for by the Fundamental theorem of algebra it should have five roots? Yes, even lowly 1 has three other roots. And the above is a way to get them.

I also straightened out my confusion on the use of the word Order. Group theory uses the term two different ways:

1 The number of elements in the group
2 The lowest power to which a group element has to be raised to return to the identity value.

The latter is actually what they officially named the Carmichael function over in number theory. The latter also roughly equates to the order of the element of a modulus ring (which we are so familiar with by now) over in number theory.

I will just chip away, concept by concept--here on groups, here on rings, here on fields, here on algebraic numbers, here on forms, here on categories--until my Frankensteinian collage starts to take form. I do not have the background to forge straight ahead. I do have a fair instinct for where to chip to get what I want, and Yes/No seldom points the right way with the back of his neck.

3. I wouldn't call anything "unnatural" Maths is just very difficult to wrap my mind round, is all, I don't have the education for it and find it too difficult to learn, though I am sure it can be learned. The Wikipedia item on cosmology expresses it well: https://en.wikipedia.org/wiki/Cosmology.

Originally Posted by desiresjab
It is above my head, too, DW. That is why I am trying to understand it. A famous mathematician once said math is an unnatural way to think. True but strange, since it seems to be the language of natural things, at the very least their superficial language. For people like Euler and Gauss it is probably not unnatural. For world class mathematicians of any era, I have no idea whether it is a natural way. I suppose it must become so after a while. Even lowly I can notice a difference in my own abilities after having stuck with math assiduously for the last year.

Some of the misconceptions I had along the way and had to correct--why, it's preposterous and laughable. Gauss or Euler or the next level down from them would never have such problems on the same material, I imagine. They invented half the stuff I am still trying to understand well. All they had to guide them were their own instincts.

* * * * *

Anyone with a cosmological thought should voice them here. This is not a math thread but only seems so at the moment. Thoughts on the subject that are not math-oriented are not interrupting anything but adding to the discussion's breadth. I do not know peoples' ages on here. I know death begins to preoccupy the mind after a certain age. Death and cosmology go together.

* * * * *

Certain questions from months ago are still haunting me. Are types of universes possible which are logically impossible to us? I don't know how one would ever answer that.

4. I remember reading a book on Teleology back in my early twenties. No, it was called The Cosmological Arguments, and had a section called the Teleological Argument. I think it also had a section called The Argument From Design. I gave philosophy the college try in those days, wading through many traditional names. It was hard wading. I did not understand a lot of it. I found Heidegger hardest of all with also the hardest name to remember how to spell. What amazed me was how men could have so many thoughts and arrive at so many fast convictions about the universe. I was wondering if I would ever arrive at one fast conviction about the joint. Lo and behold, over the last few years I finally did. You have probably heard it. Whatever there is--universe or multiverse--it cannot have come from nothing. If something exists now, this so-called nothing we came from was a false nothing, for it at least contained the possibility, the potential for something to come about, and potential is not nothing. The only thing that can come of nothing is nothing.

That does not mean that it was my thought, only that I was led there as a result of my own reflections (or so it seems) rather than lifting it from someone. That said, a million people must have said it. I said and I feel it. It is one of my few hardcore convictions about cosmology.

5. I will soon be off for a few days again, traveling to visit an ancient parent who is still fully sentient. I never take a computer. My brain is on its own when I travel, which forces me back to pen and paper if I need to calculate once I arrive, for I always drive. Driving at night on lonely highways is great for deep thinking. Just make sure no elk clips your mirror off and smashes your windshield, i.e. enter those fog banks at a crawl, honk your horn to be safer (it's lonely, right?) because elk may panic or be blinded and run right toward your headlights. They forgot already they are on a road, if they ever remembered. The noise scares them out of your way, hopefully.

* * * * *

I do not want to go away leaving misconceptions. The lattice points in WAXY in Eisenstein's rectangle in the Wiki-peja article on QR proofs represent all the ways that sets of 5 and of 3 elements can combine in pairs. Each pair is not interesting in itself, but only for which triangle it lies in, each point does not map back somewhere that tells us anything. Their whole point is that there are precisely this many of them, i.e. precisely φ/4 of them. Eisenstein never mentions φ, but what his four smaller rectangles are doing is dividing: φ(pq)/4, which can use the Euler phi function for prime rectangles, at least. Of course, he is working from the angle of Euler's criterion, because he has to make that connection and show that his diagram actually represents quadratic residues, at least in their correct number. Once we have understood him we can take the shortcut of φ/4 in perfect safety. It will work every time. However, it is faster just to multiply, but nice to know this other function we claim is ubitquitous (the phi function) offers us yet another example.

The diagonal should not be taken as another division by 2, as I mistakenly did some months past. It is a simple scalar, which can be a confusing word in mathematics. The diagram takes a ratio via the diagonal, nothing more.

I am now interested again in the idea of that limiting ratio. In other words, how much relative difference between p and q will guarantee a different number of points in the two triangles of WAXY when the Legendre symbol of the two primes is even, for we realize two 4n+3 primes must always have a different number of points, regardless of their difference or ratio. With such a small slope for its diagonal we do not expect the pair (5, 41) to produce equal numbers of points, for instance. I believe I calculated them and there were 24 and 16 points in the two triangles respectively. On the other hand we know quite well (without having actually proved it) that twin primes will always have the same number of points in both triangles. We have seen it in primes that are not twins, too. This leads us to wander what the limiting ratio is. Probably an easy question. But few questions in math are easy for me.

So anyway, that is one of the things I will be thinking about in the car, all warm and lonely. I just could not live without lonliness, or maybe would not care to anymore. It seems like a real gift bestowed through cruelty sometimes. If life ends forever, I believe that would be cruel if a conscious creator had at least as much empathy as us, and I know the thought of final ending is cruel. I think we would hope for a conscious creator with a fair bit more empathy than ourselves. Minus a conscious creator, let us hope the structure of our being is so complex and deep that it happens to include an afterlife of some kind. We understand four percent of the "stuff" in the universe so far. We have probably not penetrated our own structure even that deeply yet.

6. If the limit does not converge to one number there may be a set of numbers it converges to that would be interesting.

I agree that nothing comes from nothing. The way I look at it is that before the beginning there was no unconscious matter. There was nothing. After the beginning there must still be no unconscious matter. What we think is unconscious matter is an illusion.

Part of the problem with e raised to the pi times i equaling -1 is that we think of ex as a function graphed on an x and f(x) plane and it increases exponentially. But in that case the x is always real. I used to confuse that x-f(x) plane with the complex plane, but it is different. If one goes pi radians about a unit circle one is at the -1 point a 180 degree turn or a pi radian turn.

Math may be a deceptive way to think about reality, more than an unnatural way to think. It leads one to think that determinism and randomness are to be expected, but I don't think there is anything that is deterministic or random in the universe unless we construct it to be so, like a mathematical theory or a computer (which eventually breaks down destroying the determinism we put into the machine).

I think there are many universes since a single universe cannot be infinite without destroying the possibility of life, but they are all the same. They would follow an evolution that is similar to a spiral rather than a circle.

7. Originally Posted by YesNo
If the limit does not converge to one number there may be a set of numbers it converges to that would be interesting.

I agree that nothing comes from nothing. The way I look at it is that before the beginning there was no unconscious matter. There was nothing. After the beginning there must still be no unconscious matter. What we think is unconscious matter is an illusion.

Part of the problem with e raised to the pi times i equaling -1 is that we think of ex as a function graphed on an x and f(x) plane and it increases exponentially. But in that case the x is always real. I used to confuse that x-f(x) plane with the complex plane, but it is different. If one goes pi radians about a unit circle one is at the -1 point a 180 degree turn or a pi radian turn.

Math may be a deceptive way to think about reality, more than an unnatural way to think. It leads one to think that determinism and randomness are to be expected, but I don't think there is anything that is deterministic or random in the universe unless we construct it to be so, like a mathematical theory or a computer (which eventually breaks down destroying the determinism we put into the machine).

I think there are many universes since a single universe cannot be infinite without destroying the possibility of life, but they are all the same. They would follow an evolution that is similar to a spiral rather than a circle.
Hello. Happy to be back.

I have noticed that opinions hardly interest me anymore. Even my own cloy my thinking. I am gorged. To me an understanding of how Carmichael numbers can form ideals is worth any number of opinions or unapproachable speculations right now. I have also been wont to speculate toward much larger pictures, and will be so again. But for now my speculations must end in mathematical truth I know is there. Full understanding of consciousness is not even remotely possible at this time. But for the assiduous, Carmichael numbers are, unless you happen to plum run out of brains. I have my hopes up that I have not run out yet.

While I was away I vaguely remembered one statement from an article I read that every Carmichael number that is not already a double of a Carmichael number will have a double which is. I may have misread that statement. But I began to wonder anyway if each Carmichael and its single double form their own ideal set of which other Carmichael numbers are not members and have nothing to do with. I do not know yet. Haven't even checked yet. I was hoping you knew. I am looking for a Trojan horse in this seige.

Either Dedekind wins or I win. It has always been that way once I become obsessed. Directed obsession is the best tool a person has, it is the best one I have found. People set limits for themselves that are not necessarily true. It is a natural habit. I am interested in my own limits, the real ones, if ultimately there are any. I do not want to do IQ tests or physical puzzles, I want to see how far I can penetrate the nature of numbers. Seeing into numbers is seeing into the universe and maybe into God. Writers have a connection too, for God just spoke everything into existence according to one old text. I take the Byzantian creative ideal of Yeats as seriously as the math ideals of Dedekind. When Billy said, "The best lack all conviction, while the worst are full of passionate intensity," I do not think he meant my intensity, or his own, which aimed at understanding. Math is rarely "Modified in the guts of the living," however, as another Billy almost said, which shows the difference in the arts. Euler was as creative as Yeats. I nestle up to both.

Lunch is over. I hear the battle horn. The seige resumes.

8. Originally Posted by YesNo
It is not all n-1 residue classes that are false witnesses to make a Carmichael number. Only those that are relatively prime to n. In the case of a Carmichael number, which are squarefree, one would have to get a factor for Fermat's criterion to be accurate, but it wouldn't be accurate for an = a (mod n). That would still work.

Consider 561 = 3*11*17, a Carmichael number (assuming the python is correctly programmed):

3561 = 3 mod 561, but 3560 = 375 mod 561
11561 = 11 mod 561, but 11560 = 154 mod 361
17561 = 17 mod 561, but 17560 = 34 mod 361

There are seem to be at least three layers of tests each restricting the exponent of the witness a bit more:

an = a mod n
an-1 = 1 mod n Fermat's test
a(n-1)/2 = (a/n) mod n Euler's test
I assumed what was in red after you said it, but doesn't the statement below contradict it from the first paragraph of the linked article?

"There are composite numbers n which fail this test no matter how we choose a...."

http://www.sciencedirect.com/science...22314X07002089

9. As far as the best lacking conviction goes, it sounds like whining. But no one quotes me like they quote Yeats, which is probably a good thing.

I don't know much about Carmichael numbers except what I have explored with you. So Carmichael numbers come in pairs based on the doubling idea. That might make sense because phi(2) = 1. So Carmichael numbers can be even.

Edit: There are three tests. If one uses an = a mod n then all integers a will give the desired result for a Carmichael number a. However, if one uses an-1 = 1 mod n, then a has to be relatively prime to n for that to work. The third test should not have any Carmichael numbers although there are pseudoprimes. At least that is how I see it at the moment. I might be wrong.

10. Very good. I think you have it.

But wait! I have to have misread. How can a Carmichael number have a double when they are all odd? Scratch that bad idea. Which means I am back to wondering how something that is not a multiple can be an ideal. There was in the beginnings an ideal number theory, then there came a more general theory of ideals, which is how Carmichaels were made into ideals. At that point I believe they are talking about algebraic integers and particular behaviors in groups instead of everyday integers. The wider theory still looks like Fermat's Little Theorem with different representatives as the exponents which mean the same thing for groups as the familiar exponents mean for numbers. I will try to zero in on those group behaviors.

11. I was pleased and surprised to learn that in the ring of Polynomials all equations whose constant eqauls zero form an ideal because they all tend to zero as x vanishes toward mathematical nothingness.

Another fascinating fact was that in a ring of polynomials Carmichael numbers with only two factors occur. These are calleg Gauss-Carmichaels, or just G-Carmichaels. And I believe that left and right multiplication simply refers to the non-commutative nature of multiplication in some rings and groups. I now think when a prime splits it probably refers to the unavoidability of non commutative multiplication in the attempt to factor into prime ideals. The subtle differences between groups and numbers have to be observed. A ring of integers is not a ring of polynomials. It seems there are serious differences and serious similarities.

I think I read that the ideal also gives pros a reliable measure of how far from completely factorable a polynomial is in ideals. They add some exponents to get this value. I forget exactly which exponents though. A line of research begun by old Gauss two hundred years ago, I believe. It is ahead of myself and solid understanding. I often have to read ahead of myself.

P.S. Something like they add all the exponents not forced to zero when they impose conditions I forget. I cannot even remember the precise situation. Sometimes these half-memories do not work out, like the idea of a Carmichael double. I think I may have confused that with some kind of ideal double which indeed might exist.

12. It is interesting that Gauss-Carmichaels can have as few as two factors. I remember a few days ago being convinced that there has to be three factors in a Carmichael number in the integers. This makes me wonder why factoring in a ring of polynomials can generate something different.

13. Originally Posted by YesNo
It is interesting that Gauss-Carmichaels can have as few as two factors. I remember a few days ago being convinced that there has to be three factors in a Carmichael number in the integers. This makes me wonder why factoring in a ring of polynomials can generate something different.
It makes me wonder too. Maybe one has to get down among the greasy gears and watch this style of factorization for a while. I have never seen it done. I assume it is something different from the factoring one does in high school. How does it work? Can you factor such an expression for me?

14. Just to make sure I am not confused I looked at this article: https://en.wikipedia.org/wiki/Polynomial_ring

A polynomial ring needs some symbol X and coefficients in some ring such as the integers. Then (x+1)(x+2) = x2+ 3x + 2 is a factorization. I don't think it is anything more than that, but again, I might be missing something.

15. Originally Posted by YesNo
Just to make sure I am not confused I looked at this article: https://en.wikipedia.org/wiki/Polynomial_ring

A polynomial ring needs some symbol X and coefficients in some ring such as the integers. Then (x+1)(x+2) = x2+ 3x + 2 is a factorization. I don't think it is anything more than that, but again, I might be missing something.
I am trying to determine if that is precisely what they mean when they call a factor irreducible in a polynomial ring. It know it equates somehow to primes but is not quite a prime. I also suspect it perhaps has something to do with the roots of unity technique I observed on the abstract algebra video.

For instance, it seems to me that your example equation would not be further reducible because, for one criterion, it has already been factored to linear terms. I believe that is a strong criterion for an irreducible factor--that it be in linear terms. Even something like x4+1=0 has to be in linear terms to be factored irreducibly, I believe. I don't know if all equations can be. In fact, though, I may have read that some cannot be made irreducible even over the Complex numbers. There is much to learn along the way.

More intense study of groups has to be next. In the Wiki-peja article on Group Theory it states that Algebraic Number Theory is a special case of Group Theory, so follows the rules of the latter. This looks like Plymouth Rock to me. It means that to get my chokehold on ideals I will next have to retreat more intensely at Group Theory. I am out of sequence in my studies, but maybe I can survive it. My Abstract Algebra is ahead of my Linear Algebra (practically non existent for me) and Group theory (medicocre to half-assed decent). I seem to be learning them all at the same time, but only because it seems necessary in the quest to nail down ideals in polynimial rings.

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