[seqfan] Re: A027763 and related sequences regarding smallest k such that b^^n is not congruent to b^^(n-1) mod k, for constant b.

[israel at math.ubc.ca]

On Sep 2 2014, Wayne VanWeerthuizen wrote:


> *** Question: Is there a formal proof that all the terms in these 
> sequences must be primes or powers or primes?

Yes.  More generally, for any two integers x and y:
By the Chinese Remainder Theorem, if x != y mod k and k = k_1 * k_2 with
k_1 and k_2 relatively prime, then x != y mod k_1 or x != y mod k_2.
So the least k such that x != y mod k must be a prime or prime power.

See sequence A007978.

Cheers,
Robert