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References
edit... need improvement I think. Richard Pinch (talk) 10:44, 2 August 2008 (UTC)
- I have added [1] OEIS links and a paper by Bilu. I think this is enough for such a simple stub so I removed {{refimprove}}. PrimeHunter (talk) 13:12, 2 August 2008 (UTC)
- I have located some peer-reviewed references. I wasn't happy with just having web pages. Richard Pinch (talk) 16:20, 2 August 2008 (UTC)
Relation with Wieferich primes
editI'm thinking of proposing a merger. Richard Pinch (talk) 10:44, 2 August 2008 (UTC)
Wieferich sequence
editStart with a(1) any natural number (>1), a(n) = the smallest prime p such that (a(n-1))p-1 = 1 (mod p2) but p2 does not divide a(n-1)-1 or a(n-1)+1. 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}.
Let a(1) = 83:
- 83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}.
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)
Do values for a(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, ...
— Preceding unsigned comment added by 101.15.67.1 (talk) 14:26, 30 July 2015 (UTC)
Rename article "Double Wieferich pair"
editI suggest to rename this page to "Double Wieferich pair" or "Double Wieferich prime pair". Almost all sources dealing with these pairs call them "Double Wieferich pairs". -- Toshio Yamaguchi 10:35, 27 August 2017 (UTC)
Wieferich sequence
editShouldn't this: "the smallest prime p such that a(n-1)p-1 = 1 (mod p)" read "... = 1 (mod p^2)"? Otherwise it's true for all p. --Bur (talk) 06:08, 25 June 2022 (UTC)