Wall–Sun–Sun prime

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 V_k(p) ≡ 1 (mod p²), where V_n 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.

GeneralizationsA 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)

A Tribonacci-Wieferich prime is a prime p satisfying h(p) = h(p²), where h is the least positive integer satisfying [T_h,T_h+1,T_h+2] ≡ [T₀, T₁, T₂] (mod m) and T_n 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 P_p-1, when p congruent to 1 or 7 (mod 8), or p² divides P_p+1, when p congruent to 3 or 5 (mod 8), where P_n 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)

Wall-Sun-Sun prime aka Fibonacci-Wieferich prime, and Wall's conjecture

See also

Wieferich prime
Wolstenholme prime
Wilson prime
PrimeGrid
Fibonacci prime
Pisano period
Table of congruences

References

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.