A

prime *p* is called a Wieferich prime if 2

^{p-1}-1 is a multiple of

*p*^{2}. Only two Wieferich primes are known, 1093 and 3511, and it is suspected that only

finitely many primes share this property.

More generally, a Wieferich prime to base *b* is a prime *p* such that *p*^{2} divides *b*^{p-1}-1. Double Wieferich primes are pairs of primes *p*, *q* such that *p* is Wieferich to base *q* and vice versa.

Wieferich primes first arose in the study of Fermat's last theorem. Wieferich proved in 1909 that if a counterexample to Fermat's last theorem were to exist, the exponent of the integer powers involved must be a Wieferich prime. (Mirimanoff the following year proved that the exponent would also need to be a Wieferich prime to base 3.) More recently, double Wieferich primes have cropped up in the search for counterexamples to another famous problem, Catalan's conjecture.