Wieferich pair

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

Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).

There are only 7 Wieferich pairs known:

A Wieferich triple is a triple of prime numbers p, q and r that satisfy

There are 17 known Wieferich triples:

Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p₁, p₂, p₃, ..., p_n) such that

For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple.

For the smallest Wieferich n-tuple, see  .

Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)^p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:

The Wieferich sequence of 83:

The Wieferich sequence of 59: (this sequence needs more terms to be periodic)

However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:

The Wieferich sequence of 14:

The Wieferich sequence of 39:

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