Are there any Wall–Sun–Sun primes? If yes, is there an infinite number of them?
A k-Wall-Sun-Sun prime is defined as a prime p such that p2 divides the k-Fibonacci number (a Lucas sequence Un with (P, Q) = (k, -1)) , where is the Legendre symbol. For example, 241 is a k-Wall-Sun-Sun prime for k = 3. Thus, a prime p is a k-Wall-Sun-Sun prime iff Vk(p) ≡ 1 (mod p2), where Vn is a Lucas sequence with (P, Q) = (k, -1).
Least n-Wall-Sun-Sun prime are
- 13, 241, 2, 3, 191, 5, 2, 3, 2683, ... (start with n = 2)
In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×1014. Dorais and Klyve extended this range to 9.7×1014 without finding such a prime. In December 2011, another search was started by the PrimeGrid project. As of October 2014, PrimeGrid has extended the search limit to 2.8×1016 and continues.
Wall–Sun–Sun primes are named after Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a potential counterexample to this centuries-old conjecture.
A Tribonacci-Wieferich prime is a prime p satisfying h(p) = h(p2), where h is the least positive integer satisfying [Th,Th+1,Th+2] ≡ [T0, T1, T2] (mod m) and Tn denotes the n-th Tribonacci number. No Tribonacci-Wieferich prime exists below 1011.
A Pell-Wieferich prime is a prime p satisfying p2 divides Pp-1, when p congruent to 1 or 7 (mod 8), or p2 divides Pp+1, when p congruent to 3 or 5 (mod 8), where Pn denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell-Wieferich primes, and no others below 109. (sequence A238736 in OEIS)
- Wieferich prime
- Wolstenholme prime
- Wilson prime
- Fibonacci prime
- Pisano period
- Table of congruences
- Andrejić, V. (2006). "On Fibonacci powers" (PDF). Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A.
- Klaška, Jiří (2007), "Short remark on Fibonacci−Wieferich primes", Acta Mathematica Universitatis Ostraviensis 15 (1): 21–25.
- McIntosh, R. J.; Roettger, E. L. (2007). "A search for Fibonacci−Wieferich and Wolstenholme primes" (PDF). Mathematics of Computation 76 (260): 2087–2094. doi:10.1090/S0025-5718-07-01955-2.
- Dorais, F. G.; Klyve, D. W. (2010). "Near Wieferich primes up to 6.7 × 1015" (PDF).
- Wall–Sun–Sun Prime Search project at PrimeGrid
- Wall-Sun-Sun Prime Search statistics at PrimeGrid
- Wall, D. D. (1960), "Fibonacci Series Modulo m", American Mathematical Monthly 67 (6): 525–532, doi:10.2307/2309169
- Sun, Zhi-Hong; Sun, Zhi-Wei (1992), "Fibonacci numbers and Fermat’s last theorem" (PDF), Acta Arithmetica 60 (4): 371–388
- Klaška, Jiří (2008). "A search for Tribonacci-Wieferich primes". Acta Mathematica Universitatis Ostraviensis 16 (1): 15–20.
- Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer. p. 29. ISBN 0-387-94777-9.
- Saha, Arpan; Karthik, C. S. (2011). "A Few Equivalences of Wall-Sun-Sun Prime Conjecture". Working paper. arXiv:1102.1636.
- Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
- Weisstein, Eric W., "Wall–Sun–Sun prime", MathWorld.
- Richard McIntosh, Status of the search for Wall–Sun–Sun primes (October 2003)