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Jan 11, 2012 at 23:07 comment added François Brunault Silverman proved that the abc-conjecture implies that there are infinitely many primes $p$ such that $2^{p-1} \not\equiv 1 \pmod{p^2}$ (Wieferich's criterion and the abc-conjecture, Journal of Number Theory, 1988). This suggests that your question might be difficult...
Jan 11, 2012 at 22:34 comment added François Brunault Are some upper bounds already known for the $p$-adic valuation of the quantities you mentioned ? By comparison, I'm not aware of decent upper bounds on the $p$-adic valuation of $2^p-2$ or $(p-1)!+1$, for example.
Jan 11, 2012 at 2:38 history asked Julian Rosen CC BY-SA 3.0