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Message boards : Project Staging Area : Generalized Wieferich primes

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Profile Roger
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Joined: 27 Nov 11
Posts: 1137
ID: 120786
Credit: 266,942,391
RAC: 7,873
Found 1 prime in the 2018 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,037,982)Cullen LLR Ruby: Earned 2,000,000 credits (2,015,907)ESP LLR Ruby: Earned 2,000,000 credits (2,232,391)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,088,705)PPS LLR Ruby: Earned 2,000,000 credits (2,962,062)PSP LLR Ruby: Earned 2,000,000 credits (2,539,644)SoB LLR Ruby: Earned 2,000,000 credits (2,122,524)SR5 LLR Ruby: Earned 2,000,000 credits (2,238,295)SGS LLR Ruby: Earned 2,000,000 credits (3,560,354)TRP LLR Ruby: Earned 2,000,000 credits (2,125,391)Woodall LLR Ruby: Earned 2,000,000 credits (2,037,732)321 Sieve Turquoise: Earned 5,000,000 credits (5,190,731)Cullen/Woodall Sieve (suspended) Silver: Earned 100,000 credits (207,387)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,049,697)PPS Sieve Double Bronze: Earned 100,000,000 credits (100,422,123)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (3,227,972)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,021,659)AP 26/27 Sapphire: Earned 20,000,000 credits (20,651,644)GFN Emerald: Earned 50,000,000 credits (57,918,585)PSA Sapphire: Earned 20,000,000 credits (43,298,465)
Message 66567 - Posted: 12 Jun 2013 | 12:02:45 UTC

The definition of Wieferich prime uses base 2 for historical reasons, but mathematically there is no particular reason why we can’t consider other bases as well.
If qp(a) ≡ 0 (mod p) then ap-1 ≡ 1 (mod p2). Primes for which this is true for a = 2 are called Wieferich primes. In general they are called Wieferich primes base a. Known solutions of qp(a) ≡ 0 (mod p) for small prime values of a are:

a p 2 1093, 3511 3 11, 1006003 5 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 7 5, 491531 11 71 13 2, 863, 1747591 17 2, 3, 46021, 48947, 478225523351 19 3, 7, 13, 43, 137, 63061489 23 13, 2481757, 13703077, 15546404183, 2549536629329

http://en.wikipedia.org/wiki/Fermat_quotient#Generalized_Wieferich_primes

We've searched a = 2 up to 1x10^17. I wonder what the other bases have been searched to?
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Profile Toshio Yamaguchi
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Joined: 19 May 11
Posts: 134
ID: 99209
Credit: 665,512
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321 LLR Bronze: Earned 10,000 credits (16,424)PPS LLR Bronze: Earned 10,000 credits (10,232)SR5 LLR Bronze: Earned 10,000 credits (10,448)TRP LLR Bronze: Earned 10,000 credits (24,275)Woodall LLR Bronze: Earned 10,000 credits (19,632)PPS Sieve Bronze: Earned 10,000 credits (89,027)TRP Sieve (suspended) Bronze: Earned 10,000 credits (20,665)PSA Silver: Earned 100,000 credits (469,144)
Message 66580 - Posted: 12 Jun 2013 | 16:38:49 UTC - in response to Message 66567.

We've searched a = 2 up to 1x10^17. I wonder what the other bases have been searched to?


According to https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.pdf, Dorais and Klyve searched for Wieferich primes in bases 3, 5 and 7 from 6.5x10^9 up to 9.7x10^14 (see page 9). I guess (but haven't verified) that the ranges below 6.5x10^9 had already been searched completely, so everything up to 9.7x10^14 seems to be complete for those bases. According to http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort, all bases up to 1052 that have a prime factor up to 61 have been searched up to approximately 4.1x10^13, those with a prime factor in [67, 149] have been searched up to 2.1x10^13 and all other smaller than 1052 up to approximately 1.2x10^13.

Profile Toshio Yamaguchi
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Joined: 19 May 11
Posts: 134
ID: 99209
Credit: 665,512
RAC: 0
321 LLR Bronze: Earned 10,000 credits (16,424)PPS LLR Bronze: Earned 10,000 credits (10,232)SR5 LLR Bronze: Earned 10,000 credits (10,448)TRP LLR Bronze: Earned 10,000 credits (24,275)Woodall LLR Bronze: Earned 10,000 credits (19,632)PPS Sieve Bronze: Earned 10,000 credits (89,027)TRP Sieve (suspended) Bronze: Earned 10,000 credits (20,665)PSA Silver: Earned 100,000 credits (469,144)
Message 66602 - Posted: 13 Jun 2013 | 10:18:43 UTC

See also the paper by Montgomery available here.

Message boards : Project Staging Area : Generalized Wieferich primes

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