Fermat's little theorem states that a p&1 #1(mod p), if p is a prime number and a is an integer coprime with p. The primes for which a p&1 #1(mod p 2 ) are sometimes called base-a Wieferich primes, or simply Wieferich primes if a=2. This stems from a theorem of Wieferich that asserts that the first case of Fermat's last theorem holds if p is not a Wieferich prime (similar results were obtained for a few other bases). This is of historical interest only now, but Wieferich primes appear in other contexts, most notably they are the primes for which ``the derivative of a vanishes,'' when one pursues a number field-function field analogy, such as in ([Bu], [I], [Sm]). Numerical evidence indicates there are very few Wieferich primes for a given base, for instance p=1093, 3511 are the only (base-2) Wieferich primes less than 10 12 ([CDP]). A naive heuristic argument suggests that there should be about log log x Wieferich primes up to x, whereas the analogy with function fields would suggest there are only finitely many Wieferich primes. These two possibilities (and an unlikely third) are contrasted in [I]. However there are very few unconditional results on Wieferich primes. For instance it is not known for any base a if there are infinitely many non-Wieferich primes to base a, although Silverman showed that this follows from the abc-conjecture and Johnson [J] showed that Mersenne primes are not (base-2) Wieferich primes. About the only unconditional result is due to Granville [G] (see also Puccioni [P]) and asserts that if a is prime (this condition may be removed. as we shall see) and a p&1 # 1(mod p 2 ) for all sufficiently large primes p, then a p&1 #1(mod p 3 ) for infinitely many primes. Replacing the multiplicative group by the group of an elliptic curve one obtains an analogous notion of elliptic Wieferich primes, introduced by Silverman [S], where it is shown that the abcconjecture implies the existence of infinitely many nonelliptic Wieferich primes for some special elliptic curves.
The purpose of this note is to investigate further the elliptic Wieferich primes, we prove an analogue of Granville's result in the number field case