Wall–Sun–Sun prime

Wall–Sun–Sun prime
Named after	Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun
Publication year	1992
Number of known terms	0
Conjectured number of terms	Infinite

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 ≠ 2, 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 $\textstyle\left(\frac{p}{5}\right)$ has the values

\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 if L_p ≡ 1 (mod p²), where L_p is the p-th Lucas number.^[1]^:42

Existence

Unsolved problem in mathematics:

Are there any Wall–Sun–Sun primes? If yes, are there an infinite number of them?
(more unsolved problems in mathematics)

Originally, Donald Dines Wall hypothesized the non-existence of Fibonacci Wieferich primes, but could not prove they were impossible, hence the question remains open. It has since 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 k-Wall–Sun–Sun prime is defined as a prime p such that p² divides the k-Fibonacci number (a Lucas sequence U_n with (P, Q) = (k, −1)) $F_k\left(p - \left(\frac{k^2+4}{p}\right)\right)$ , 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)

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) In fact, Pell–Wieferich primes are 2-Wall–Sun–Sun primes.

Near-Wall–Sun–Sun primes

A prime p such that $F_{p - \left(\frac{p}{5}\right)} \equiv Ap \pmod{p^2}$ with small |A| is called near-Wall–Sun–Sun prime.^[10] Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes.