A Wieferich prime is a prime p which is a solution to the congruence equation 2^(p - 1) congruent 1 (mod p^2). Note the similarity of this expression to the special case of Fermat's little theorem 2^(p - 1) congruent 1 (mod p), which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220), with none other less than 4×10^12 , a limit since increased to 1.25×10^15 and subsequently to 4.96854×10^17 by PrimeGrid as of November 2015.

abc conjecture | double Wieferich prime pair | Fermat quotient | Fermat's last theorem | integer sequence primes | Mersenne number | Mirimanoff's congruence | powerful number