Here's something on decimal expansions. I have only read the first page: http://people.csail.mit.edu/kuat/cou...expansions.pdf
Continued fractions also have repetitions that might be useful.
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Here's something on decimal expansions. I have only read the first page: http://people.csail.mit.edu/kuat/cou...expansions.pdf
Continued fractions also have repetitions that might be useful.
I found that a pretty tough paper. One expects no less at the graduate level. On the other hand I see most of what is going on in it. Primitive roots mod (p) appear to have periods p-1, if I interpreted it correctly. Of course we already knew that about primitive roots, at least what their period would be, powered up. The good part is that behavior carries over to the decimal expansion.Quote:
Originally Posted by YesNo
I haven't had time to finish it. I am wondering how they will prove the main result on the first page that given an integer d and a base a (not equal to 2) there is a prime p such that the length of the period of the expansion of 1/p is d. I would not think this would work for any d and it doesn't seem to work for base 2. Supposedly there is no prime p that has period of length 6 in base 2.
I guess it is a curious fact:
(6k+1)(12k+1)(18k+1) may be a Carmichael number whenever all three factors are prime, is how I took it. Is it sufficient but not necessary? The curious fact is that 561 is not of this form, for it is (3)(11)(17). There must be more than one breed of Carmichael number. It goes to show that in the deep structure of numbers things are never quite simple. I may never know why a Carmichael number just had to be, or whether in the time of Euler and Gauss it was highly suspected there were such beasts. I do not see 561 defeating either of that pair's calculating ability, so perhaps they were not too suspicious in their day. Was Fermat's little theorem considered a good primality test then? I do not see either of them falling for that, especially Gauss, who only showed winning hands. It is kind of a nice historical question apart from understanding the technically difficult parts.
As I look at the complex formulas for Carmichael numbers and repetends, I see familiar friends like the G.C.D and the phi function filling them out. Sometimes there is that big O term you were talking about, the error term, which comes last and expresses the probablility of error. I think Hardy and Littlewood made a formula for the density of Carmichael numbers, but it could be another formula I am confusing it with.
The link you provided for the repetends study was pretty deep. I did not like their style. I think by going over it quite a few times I can extract most of it. I might find some other articles dealing with the same subject.
I am going to need some work on why i and e and pi are in those formulas, in fact I have noticed they seem to be a staple in many of the higher number theoretic formulas I am seeing. I have to get comfortable with that. Gauss's criterion for recognizing a mathematician of the first class in the making was immediate understanding of Euler's formula eiπ+1=0. I do not qualify. I have to find out why these terms have found a home in the number theoretic formulas. I believe it must be through the complex number system, and I know some trig identities are involved.
Oh boy! I think I found the key statement I was looking for. It was in a Wiki-peja article.
The length of the repetend of 1/p is equal to the order of 10 (mod p). If 10 is a primitive root mod p, the reptend length is equal to p-1; if not, the repetend length is a factor of p-1.
Just how precisely those fractions which produce lengths which are factors of p-1 can be nailed down, I am not sure. It may be something found in the link you provided for repetends. But that factor which is the length of the repetend is merely the order of 10 (mod p), right? That says it all. Those are the bare facts, the rest is just proofs.
That sounds like a good distinction. I don't understand why the base needs to be a primitive root mod p, but it should have something to do with it.
I think there are Carmichael numbers with more than three prime factors, but not less than three.
One doesn't have to understand everything in an article, nor even read all of it. I rarely finish reading things.
The i, e and pi are in the cyclotomic polynomials to get roots of xn - 1 in that article. I think that is just one way to get uniformly spaced points around the unit circle. One could use sines and cosines, but this is more compact. https://en.wikipedia.org/wiki/Euler's_identity I don't have an intuitive feel for this either. It is just a way to calculate. It is kind of like quantum physics. One can calculate and get useful results without knowing what it is one is talking about.
I believe because anything but a primitive root will power up to 1 (mod p) before its p-1 power. The power at which it reaches 1 for the first time is the order. The order of 10 (mod p) is the length of the repetend. Of course 10p-1 is not exactly easy to calculate for large primes. After one had gone as far as (p-1)/2, the largest possible factor of p-1 except for itself, one could safely conclude they were dealing with a primitive root (mod p).Quote:
Originally Posted by YesNo
Yes, I see that now. The primitive root will be able to generate all relatively prime values less than the prime p and there are p-1 of them.
What I do not see is the structural similarity between various types of Carmichaels. That (6k+1)(12k+1)(18k+1) criterion was of course for a three-factor Carmichael. But that three-factor Carmichael (3)(11)(17) is not of the above form, and I have no idea whether an infinite number of Carmichael numbers are of a different form than the one above. So what form is it? I have not located the unifying principal between all Carmichael numbers. There are infinitely many of them, and even infinitely many Carmichaels with any number of factors you care to name. It seems to me there has to be some principle unifying all Carmichael numbers. It is probably sitting right in front of my nose and I cannot see it. I will see it, at least I think so presently.
This unifying principle between all Carmichael numbers of any form, may be what the Chinese civil servant recently tapped into. I feel it is tappable. I feel you and I have to tap it now, since we went ahead and challenged it, asking our not so innocent questions, and we are sure to be named wusses if we back away now without the answer. We did not back away from quadratic reciprocity until we knew there was at that time no more getting to be had from it with the tools we were using. I dread these prolonged struggles because I am lazy and always look for an easy way, in slovenly accordance with Occam's razor.
It would certainly be nice if 561 were a single rogue example and all other Carmichaels were of the above form. I highly doubt that, but do not yet know that it is untrue. I have a heuristic theory of rogue solutions underway right now which I hope to present ri'cheer in the near future.
A more careful reading of Carmichaels reveals that (6k+1)(12k+1)(18k+1) produces a subset of Carmichaels when all three factors are prime. This was proved in 1939. It is not yet proven that this is an infinitely repeatable subset, but highly suspected.
The fact that there are infinitely many Carmichael ideals sounds a bit prohibitive for finding one nature that defines them all. I do not know enough about the theory of ideals yet. There may not be an approach to them that is not laden with abstract algebraic notations. Perhaps we should proceed as if there were one nature to be found, until otherwise is shown which we can recognize.
And here is a highly provocative statement from Wiki-peja:
Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in OK. For example, if p is any prime number that is 1 mod 4, the ideal (p) in the Gaussian integers Z[i] is a Carmichael ideal.
Which Wikipedia article were you reading?
I l,ooked through the stuff I read. Could not find the quotes I gave. It was one of the following links. The one on category theory is killer abstract. It is hard to find a good entry point for any one of these subjects because they are all inter-related and used in the definitions of each other.Quote:
Originally Posted by YesNo
https://en.wikipedia.org/wiki/Category_theory
https://en.wikipedia.org/wiki/Injective_function
https://en.wikipedia.org/wiki/Ideal_class_group
https://en.wikipedia.org/wiki/Carmichael_number
https://en.wikipedia.org/wiki/Ideal_(ring_theory)
https://en.wikipedia.org/wiki/Abstract_algebra
https://en.wikipedia.org/wiki/Quotient_ring
https://en.wikipedia.org/wiki/Fundam..._homomorphisms
https://en.wikipedia.org/wiki/Surjective_function
Those articles like a good starting point. I haven't read read all, but I will start with the one on Carmichael numbers.