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Welcome to the Wieferich and Wall-Sun-Sun Prime Search
A Wall–Sun–Sun (or Fibonacci–Wieferich) prime is a prime p > 5 in which p^2 divides the Fibonacci number  , where the Legendre symbol  is defined as 
They are named after Donald Dines Wall and twin brothers Zhi-Hong Sun and Zhi-Wei Sun. Drawing on Wall's work, in 1992 the brothers proved that if the first case of Fermat's last theorem was false for a certain prime p, then that p would have to be a Wall–Sun–Sun prime.
Although it has been conjectured that infinitely many exist, there are no known Wall–Sun–Sun primes. As of December 2011, if any exist, they must be > 9.7e14. The PRPNet search began here, and the BOINC search will as well for double-checking purposes.
The lack of success has lead to an interest in "Near" Wall-Sun_Sun primes. They are defined as special instances (with small |A|) of F_(p-(p/5)) = Ap (mod p^2).
A prime p is a Wieferich prime if p^2 divides 2^(p-1) - 1. They are named after Arthur Wieferich who in 1909 proved that if the first case of Fermat’s last theorem is false for the exponent p, then p satisfies the criteria a^(p-1) = 1 (mod p^2) for a=2.
Notice the similarity in the expression p^2 divides 2^(p-1) - 1 to the special case of Fermat's little theorem p divides 2^(p-1) - 1.
Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511. The rarity of these primes has lead to an interest in "Near" Wieferich primes. They are defined as special instances (with small |A|) of 2^((p−1)/2) ≡ ±1 + Ap (mod p^2).
Search History
Wall-Sun-Sun
Search limit Author Year
1e9 Williams 1982
2^32 Montgomery 1991
1e14 Knauer and McIntosh 2003
2e14 McIntosh and Roettger 2005
9.7e14 Dorais and Klyve 2011
10e14 PrimeGrid 2011-12-28
15e14 PrimeGrid 2012-01-10
20e14 PrimeGrid 2012-01-22
25e14 PrimeGrid 2012-03-02
60e14 PrimeGrid 2012-07-29
28e15 PrimeGrid 2014-03-31
Wieferich
Search limit Author Year
16000 Beeger 1940
50000 Froberg unknown
100000 Kravitz 1960
200183 Pearson 1964
500000 Riesel 1964
3e7 Froberg 1968
3e9 Brillhart, Tonascia, and Weinberger 1971
6e9 Lehmer 1981
6.1e10 Clark 1996
4e12 Crandall, Dilcher, and Pomerance 1997
4.6e13 Brown and McIntosh 2001
2e14 Crump 2002
1.25e15 Knauer and Richstein 2005
3e15 Carlisle, Crandall, and Rodenkirch 2006
6.7e15 Dorais and Klyve 2011
10e15 PrimeGrid 2012-01-13
14e15 PrimeGrid 2012-04-14
14e16 PrimeGrid 2014-08-11
Note that the PrimeGrid searches are not double checked yet.
Classical Definition of nearness
A prime p satisfying the congruence F_(p-(p/5)) ≡ Ap (mod p^2) with small |A| is commonly called a near-Wall-Sun-Sun prime. A prime p satisfying the congruence 2^((p−1)/2) ≡ ±1 + Ap (mod p^2) with small |A| is commonly called a near-Wieferich prime. Therefore, we are going to classify finds as follows:
Wall-Sun-Sun or Wieferich prime: A = 0
|A| <= 10
|A| <= 100
|A| <= 1000
Additional Information
- Wall-Sun-Sun: The Prime Glossary
- Wall-Sun-Sun: Math World
- Wall-Sun-Sun: Wikipedia
- McIntosh, R. J. (2004), Wall-Sun-Sun (Fibonacci Wieferich) Search Status e-mail to Paul Zimmermann.
- Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
- McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes", Math. Comp. 76 (260):2087–2094.
- Wieferich: The Prime Glossary
- Wieferich: Math World
- Wieferich: Wikipedia
- Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
- Knauer, Joshua; Richstein, Jörg (2005), "The continuing search for Wieferich primes", Math. Comp. 74 (251):1559–1563.
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