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.
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$\begingroup$ 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. :) $\endgroup$– fedjaCommented Nov 12, 2012 at 14:52
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1$\begingroup$ 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 $p^N \mid (2^{p-1}-1)$?) $\endgroup$– Greg MartinCommented Nov 12, 2012 at 20:05
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I don't think this is known. You might want to have a look at:
A. Granville, Refining the conditions on the Fermat quotient, Mathematical Proceedings of the Cambridge Philosophical Society, 98 (1985) 5-8.
You might be able to to say more assuming the ABC conjecture, which may or may not have been proved.
answered Nov 12, 2012 at 13:37