Wieferich

Wieferich refers to a concept in number theory and to the German mathematician associated with it. In this context, a Wieferich prime (base a) is a prime p such that a^(p-1) ≡ 1 modulo p^2; equivalently, p^2 divides a^(p-1) − 1. The case a = 2 is the most studied; primes p with 2^(p-1) ≡ 1 (mod p^2) are called base-2 Wieferich primes.

The term is named after Arthur Wieferich, who studied related congruences in the early 20th century. The notion highlights departures from Fermat's little theorem in a stronger modulo setting and is connected to the study of Fermat quotients and p-adic properties of bases.

As of contemporary results, only a few base-2 Wieferich primes are known, with 1093 and 3511 being the smallest examples. No others have been found below very large computational bounds, and it remains unknown whether infinitely many base-2 Wieferich primes exist. For other bases a, Wieferich primes are even rarer and less thoroughly understood, and their existence beyond the known examples is an open area of research.

Why they matter: Wieferich primes are a natural object in the arithmetic of modular powers and have implications for certain primality tests and cryptographic considerations that depend on congruences modulo higher powers of p. The topic continues to attract interest for its blend of computational search and deep number-theoretic structure.