Wieferich pair

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

p^{q − 1} ≡ 1 (mod q²) and q^{p − 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 of Mihăilescu's theorem (formerly known as Catalan's conjecture).

Known Wieferich pairs

There are only 7 Wieferich pairs known:

(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequences  A124121,  A124122 and  A126432 in OEIS)

Wieferich triple

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

p^{q − 1} ≡ 1 (mod q²), q^{r − 1} ≡ 1 (mod r²), and r^{p − 1} ≡ 1 (mod p²).

There are 9 known Wieferich triples:

(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (5, 20771, 18043), (5, 53471161, 193), (5, 6692367337, 1601), (13, 863, 23), (83, 13691, 821), and (1657, 2281, 1667). (sequences  A253683,  A253684 and  A253685 in OEIS)

Wieferich sequence

Start with a(1), it can be any natural number (>1), 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 natural number a(1) makes this sequence become periodic, for example, let a(1) = 2:

2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (a Wieferich triple)

Let a(1) = 83:

83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. (a Wieferich pair)

Let a(1) = 59 (a longer sequence):

59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ... it also gets 5.

However, there are many values of a(1) with unknown status, for example, let a(1) = 3:

3, 11, 71, 47, ? (There are no known Wieferich primes in base 47).

Let a(1) = 14:

14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 2² = 4 divides 29 - 1 = 28)

Let a(1) = 39 (a longer sequence):

39, 8039, 617, 101, 1050139, 29, ? (It also gets 29)

It is unknown that values for a(1) > 1 exist such that the resulting sequence does not eventually become periodic.

When a(n - 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)

See also

• Wieferich prime
• Fermat quotient

Further reading

• 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).