Quote:
Originally Posted by carpetpool
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 = (r1)/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
or
(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
if
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).