# Wieferich pair

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

*p*^{q − 1}≡ 1 (mod*q*^{2}) and*q*^{p − 1}≡ 1 (mod*p*^{2})

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]}

## Contents

## Known Wieferich pairs[edit]

There are only 7 Wieferich pairs known:^{[3]}^{[4]}

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

## Wieferich triple[edit]

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

*p*^{q − 1}≡ 1 (mod*q*^{2}),*q*^{r − 1}≡ 1 (mod*r*^{2}), and*r*^{p − 1}≡ 1 (mod*p*^{2}).

There are 17 known Wieferich triples:

- (2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 3188089), (5, 20771, 18043), (5, 20771, 950507), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (41, 138200401, 2953), (83, 13691, 821), (199, 1843757, 2251), (431, 2393, 54787), and (1657, 2281, 1667). (sequences A253683, A253684 and A253685 in OEIS)

## Barker sequence[edit]

**Barker sequence** or **Wieferich n-tuple** is a generalization of Wieferich pair and Wieferich triple. It is primes (

*p*

_{1},

*p*

_{2},

*p*

_{3}, ...,

*p*

_{n}) such that

*p*_{1}^{p2 − 1}≡ 1 (mod*p*_{2}^{2}),*p*_{2}^{p3 − 1}≡ 1 (mod*p*_{3}^{2}),*p*_{3}^{p4 − 1}≡ 1 (mod*p*_{4}^{2}), ...,*p*_{n−1}^{pn − 1}≡ 1 (mod*p*_{n}^{2}),*p*_{n}^{p1 − 1}≡ 1 (mod*p*_{1}^{2}).^{[5]}

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 A271100.

## Wieferich sequence[edit]

**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:

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

The Wieferich sequence of 83:

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

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

- 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, the Wieferich sequence of 3:

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

The Wieferich sequence of 14:

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

The Wieferich sequence of 39:

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

It is unknown that values for *k* exist such that the Wieferich sequence of *k* does not become periodic. Eventually, it is unknown that values for *k* exist such that the Wieferich sequence of *k* is finite.

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[edit]

## References[edit]

**^**Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture".*J. Reine Angew. Math.***572**: 167–195. doi:10.1515/crll.2004.048. 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).**^**List of all known Barker sequence

## Further reading[edit]

- Bilu, Yuri F. (2004). "Catalan's conjecture (after Mihăilescu)".
*Astérisque*.**294**: vii, 1–26. Zbl 1094.11014. - Ernvall, Reijo; Metsänkylä, Tauno (1997). "On the
*p*-divisibility of Fermat quotients".*Math. Comp.***66**(219): 1353–1365. doi:10.1090/S0025-5718-97-00843-0. MR 1408373. Zbl 0903.11002. - Steiner, Ray (1998). "Class number bounds and Catalan's equation".
*Math. Comp*.**67**(223): 1317–1322. doi:10.1090/S0025-5718-98-00966-1. MR 1468945. Zbl 0897.11009.