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
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
A124121, A124122 and A126432 in OEIS)
See also
References
- ↑ Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 572: 167–195. MR 2076124.
- ↑ Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
- ↑ Weisstein, Eric W., "Double Wieferich Prime Pair", MathWorld.
- ↑ A124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).
Further reading
- Yuri Bilu (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque 294: vii, 1–26.
- R. Ernvall; T. Metsänkylä (1997). "On the p-divisibility of Fermat quotients". Math. Comp. 66 (219): 1353–1365. doi:10.1090/S0025-5718-97-00843-0.
- Ray Steiner (1998). "Class number bounds and Catalan's equation". Math. Comp. 67 (213): 1317–1322. doi:10.1090/S0025-5718-98-00966-1.
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