Wieferich pair

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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[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). (sequences OEISA124121, OEISA124122 and OEISA126432 in OEIS)

Wieferich triple[edit]

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 13 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), (41, 138200401, 2953), (83, 13691, 821), and (1657, 2281, 1667). (sequences OEISA253683, OEISA253684 and OEISA253685 in OEIS)

Barker sequence[edit]

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

p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12).[5]

For example, (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 OEISA271100.

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 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)

See also[edit]

References[edit]

  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. ^ OEISA124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5).
  5. ^ [1]

Further reading[edit]