Quote:
Originally Posted by Dr Sardonicus

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.