A prime p is called a Wieferich prime if 2p-1-1 is a multiple of p2. 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 p2 divides bp-1-1. Double Wieferich primes are pairs of primes pq 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.

Log in or registerto write something here or to contact authors.