A pair of prime numbers (p, q) such that p^(q - 1) congruent 1 (mod q^2) and q^(p - 1) congruent 1 (mod p^2). The only known examples are (2, 1093), (3, 1006003), (5 , 1645333507), (83, 4871), (911, 318917), and (2903, 18787). If the equation of Catalan's Diophantine problem x^p - y^q = ± 1 has a nontrivial solution in integers x, y and primes p, q greater than 3, then (p, q) must be a double Wieferich pair, as proved in 2000 by Mihailescu.

Catalan's conjecture | Wieferich prime