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 Germany 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). (sequence OEIS: A124121 and OEIS: A124122 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 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 OEIS: A253683, OEIS: A253684 and OEIS: 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, ..., 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, (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: A271100, for the ordered set of all Wieferich tuples, see OEIS: 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)
See also
References
- ↑ Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 2004 (572): 167–195. doi:10.1515/crll.2004.048.
- ↑ Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture.
- ↑ Weisstein, Eric W.. "Double Wieferich Prime Pair". http://mathworld.wolfram.com/DoubleWieferichPrimePair.html.
- ↑ OEIS: 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
- Bilu, Yuri F. (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque 294: vii, 1–26.
- 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. Bibcode: 1997MaCom..66.1353E.
- Steiner, Ray (1998). "Class number bounds and Catalan's equation". Math. Comp. 67 (223): 1317–1322. doi:10.1090/S0025-5718-98-00966-1. Bibcode: 1998MaCom..67.1317S.
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