Wieferich pair

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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. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof[1] of Mihăilescu's theorem (formerly known as Catalan's conjecture).[2]

Known Wieferich pairs[edit]

There are only seven Wieferich pairs known:[3][4]

(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequences OEISA124121, OEISA124122 and OEISA126432 in OEIS)

See also[edit]

References[edit]

  1. ^ Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 572: 167–195. MR 2076124. 
  2. ^ Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
  3. ^ Weisstein, Eric W., "Double Wieferich Prime Pair", MathWorld.
  4. ^ OEISA124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).

Further reading[edit]