Wieferich pair

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 7 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)

Wieferich triple

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

pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2).

There are 12 known Wieferich triples:

(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (5, 20771, 18043), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (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

References

  1. ↑ Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 572: 167–195. MR 2076124.
  2. ↑ Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
  3. ↑ Weisstein, Eric W., "Double Wieferich Prime Pair", MathWorld.
  4. ↑  A124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).

Further reading

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