# multiplicative order of 2 mod p^N

Does there exist a positive integer N such that for all odd primes p the multiplicative order of 2 mod p is strictly less than the multiplicative order of 2 mod p^N ? Once again, references would be greatly appreciated as this pertains to ongoing graduate research.

• Since only two Wieferich primes are known, there is hardly any chance anyone will be able to say "No". It is a bit harder to quote something equally convincing to demonstrate that the probability of a qualified "Yes" answer is also $0$, but rest assured that if the graduate student in question can figure it out, he'll have no problems with competition either for priority or on the job market. In other words, we are here to tell you what we know, not to do the impossible. You have to accomplish the latter by yourself. :) – fedja Nov 12 '12 at 14:52
• Just wanted to point out that the following is an equivalent formulation: Does there exist a positive integer $N$ such that there does not exist a prime $p$ for which $p^N$ divides $2^{p-1}-1$? (Or, is it possible that for all $N\ge1$, there exists a prime $p$ such that $p^N \mid (2^{p-1}-1)$?) – Greg Martin Nov 12 '12 at 20:05