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Old 2020-02-15, 20:06   #6
carpetpool's Avatar
Nov 2016

2·3·53 Posts

Originally Posted by carpetpool View Post
Yes, it should be false in general (Roman B. Popovych's conjecture could be true however).

Assuming the falsehood of the original conjecture, what I was trying to illustrate is that counterexamples should have very specific properties, namely that the only counterexamples (with r prime) should be Carmichael numbers of order(m) where m = (r-1)/2. There is no proof (that I know of), but there is evidence that this should be true.

There should also exist upper bounds on either the Carmichael numbers of order(m) < x (and show that it is 0 for some x), or there is a lower bound such that the smallest Carmichael numbers of order(m) > L.

If these two arguments are true, then there should exist a modified version of the conjecture that is true as we will have conditions for which n must be prime.
I can't seem to get away with it

(x - 1)^(t*n + n) = x^n - 1 mod(n, x^r - 1) where t = 2*r*(n^((r - 1)/2) - 1) if jacobi(n | r) = -1
(x - 1)^(t*n + n) = x^n - 1 mod(n, x^r - 1) where t = (n^((r - 1)/2) - 1) if jacobi(n | r) = 1
n^2 ≠ 1 mod r and r > 5.

As there are infinitely many values of w such that (x - 1)^w = x^n - 1 mod(n, x^r - 1), proving that w = t*n + n is the smallest such integer (apart from n itself) is false:

n = 61, r = 7

So now I seem to have dug a deeper hole into solving this problem. We need a lower bound B such that
(x - 1)^w ≠ x^n - 1 mod(n, x^r - 1)

if w is any integer (n < w < B).
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