# Wall–Sun–Sun prime

 Are there any Wall–Sun–Sun primes? If yes, is there an infinite number of them?

In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.

## Definition

A prime p > 5 is called a Wall–Sun–Sun prime if p2 divides the Fibonacci number $F_{p - \left(\frac{{p}}{{5}}\right)}$, where the Legendre symbol $\textstyle\left(\frac{{p}}{{5}}\right)$ is defined as

$\left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5\\ -1 &\text{if }p \equiv \pm2 \pmod 5 \end{cases}$

Equivalently, a prime p is a Wall–Sun–Sun prime iff Lp ≡ 1 (mod p2), where Lp is the p-th Lucas number.[1]:42

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)) $F_k({p - \left(\frac{{k^2+4}}{{p}}\right)})$, where $\left(\frac{{k^2+4}}{{p}}\right)$ 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)

## Existence

It has been conjectured that there are infinitely many Wall–Sun–Sun primes.[2] No Wall–Sun–Sun primes are known as of October 2014.

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×1014.[3] Dorais and Klyve extended this range to 9.7×1014 without finding such a prime.[4] In December 2011, another search was started by the PrimeGrid project.[5] As of October 2014, PrimeGrid has extended the search limit to 2.8×1016 and continues.[6]

## History

Wall–Sun–Sun primes are named after Donald Dines Wall,[7] 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.[8] 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.

## Generalizations

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.[9]

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)

## References

1. ^ Andrejić, V. (2006). "On Fibonacci powers" (PDF). Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A.
2. ^ Klaška, Jiří (2007), "Short remark on Fibonacci−Wieferich primes", Acta Mathematica Universitatis Ostraviensis 15 (1): 21–25.
3. ^ 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.
4. ^ Dorais, F. G.; Klyve, D. W. (2010). "Near Wieferich primes up to 6.7 × 1015" (PDF).
5. ^ Wall–Sun–Sun Prime Search project at PrimeGrid
6. ^ Wall-Sun-Sun Prime Search statistics at PrimeGrid
7. ^ Wall, D. D. (1960), "Fibonacci Series Modulo m", American Mathematical Monthly 67 (6): 525–532, doi:10.2307/2309169
8. ^ Sun, Zhi-Hong; Sun, Zhi-Wei (1992), "Fibonacci numbers and Fermat’s last theorem" (PDF), Acta Arithmetica 60 (4): 371–388
9. ^ Klaška, Jiří (2008). "A search for Tribonacci-Wieferich primes". Acta Mathematica Universitatis Ostraviensis 16 (1): 15–20.