Wieferich pair

Wieferich pair

In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy

pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2)

Wieferich pairs are named after German mathematician Arthur Wieferich.

The only known Wieferich pairs (sequence and in OEIS) are (2, 1093), (3, 1006003), (5, 1645333507), (83, 4871), (911, 318917), and (2903, 18787).

Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).

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