Wall–Sun–Sun prime

Definition

The period of Fibonacci numbers F n modulo prime p is called the Pisano period and denoted π ( p ) . It follows that p divides F π ( p ) . A prime p such that p² divides F π ( p ) is called a Wall–Sun–Sun prime.

For a prime p ≠ 2, 5, the Pisano period π ( p ) is known to divide p − ( p 5 ) , where the Legendre symbol ( p 5 ) has the values

( p 5 ) = { 1 if  p ≡ ± 1 ( mod 5 ) ; − 1 if  p ≡ ± 2 ( mod 5 ) .

This observation gives rise to an equivalent definition that a prime p is a Wall–Sun–Sun prime if p² divides the Fibonacci number F p − ( p 5 ) .

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.

Existence

In a study of the Pisano period k ( p ) , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than 10000 . He writes:

The most perplexing problem we have met in this study concerns the hypothesis k ( p 2 ) ≠ k ( p ) . We have run a test on digital computer which shows that k ( p 2 ) ≠ k ( p ) for all p up to 10000 ; however, we cannot prove that k ( p 2 ) = k ( p ) is impossible. The question is closely related to another one, "can a number x have the same order mod p and mod p 2 ?", for which rare cases give an affirmative answer (e.g., x = 3 , p = 11 ; x = 2 , p = 1093 ); hence, one might conjecture that equality may hold for some exceptional p .

It has since been conjectured that there are infinitely many Wall–Sun–Sun primes. No Wall–Sun–Sun primes are known as of April 2016.

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 April 2016, PrimeGrid has extended the search limit to 1.9×10¹⁷ and continues.

History

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.

Generalizations

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.

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

Near-Wall–Sun–Sun primes

A prime p such that F p − ( p 5 ) ≡ A p ( mod p 2 ) with small |A| is called near-Wall–Sun–Sun prime. Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes.

Wall–Sun–Sun primes with discriminant D

Wall–Sun–Sun primes can be considered in the field Q D with discriminant D. For the conventional Wall–Sun–Sun primes, D = 5. In the general case, a Lucas–Wieferich prime p associated with (P, Q) is a Wieferich prime to base Q and a Wall–Sun–Sun prime with discriminant D = P² – 4Q. In this definition, the prime p should be odd and not divide D.

It is conjectured that for every natural number D, there are infinitely many Wall–Sun–Sun primes with discriminant D.

The case of ( P , Q ) = ( k , − 1 ) corresponds to the k-Wall–Sun–Sun primes, for which Wall–Sun–Sun primes represent a special case with k = 1. The k-Wall–Sun–Sun primes can be explicitly defined as primes p such that p² divides the k-Fibonacci number F k ( π k ( p ) ) , where F_k(n) = U_n(k, −1) is a Lucas sequence of first kind with discriminant D = k² + 4 and π k ( p ) is the Pisano period of k-Fibonacci numbers modulo p. For a prime p ≠ 2 and not dividing D, this condition is equivalent to any of the following two:

p² divides F k ( p − ( D p ) ) , where ( D p ) is the Kronecker symbol;

V_p(k, −1) ≡ k (mod p²), where V_n(k, −1) is a Lucas sequence of the second kind.

The smallest k-Wall–Sun–Sun prime for k = 2, 3, ... are

13, 241, 2, 3, 191, 5, 2, 3, 2683, ... (sequence A271782 in the OEIS)