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{{cite journal | author=Preda Mihăilescu | authorlink=Preda Mihăilescu | title=Primary Cyclotomic Units and a Proof of Catalan's Conjecture | journal=J. Reine Angew. Math. | volume=2004 | issue=572 | year=2004 | pages=167–195 |mr=2076124 | doi=10.1515/crll.2004.048}} of Mihăilescu's theorem (formerly known as Catalan's conjecture).Jeanine Daems [http://www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf A Cyclotomic Proof of Catalan's Conjecture].

Known Wieferich pairs

There are only 7 Wieferich pairs known:{{MathWorld|title=Double Wieferich Prime Pair|urlname=DoubleWieferichPrimePair}}{{OEIS2C|id=A124121}}, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).

:(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence {{OEIS2C|id=A124121}} and {{OEIS2C|id=A124122}} 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 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 {{OEIS2C|id=A253683}}, {{OEIS2C|id=A253684}} and {{OEIS2C|id=A253685}} in OEIS)

Barker sequence

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

:p₁^{p₂ − 1} ≡ 1 (mod p₂²), p₂^{p₃ − 1} ≡ 1 (mod p₃²), p₃^{p₄ − 1} ≡ 1 (mod p₄²), ..., p_n−1^{p_n − 1} ≡ 1 (mod p_n²), p_n^{p₁ − 1} ≡ 1 (mod p₁²).[http://www.cecm.sfu.ca/~mjm/WieferichBarker/Data/AllCycles.txt List of all known Barker sequence]

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 {{oeis|id=A271100}}, for the ordered set of all Wieferich tuples, see {{oeis|id=A317721}}.

Wieferich sequence

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 primes in base 29 except 2, but 2² = 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)

Wieferich pair

Known Wieferich pairs

Wieferich triple

Barker sequence

Wieferich sequence

See also

References

Further reading