In mathematics, a Wall-Sun-Sun prime is a certain kind of prime number. A prime p > 5 is called a Wall-Sun-Sun prime if p² divides

F(p − (p|5)),

where F(n) is the n-th Fibonacci number and (a|b) is the Legendre symbol of a and b.

Wall-Sun-Sun primes are named after D. D. 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 counterexample to this century-old conjecture.

No Wall-Sun-Sun primes are known to date; if any exist, they must be > 10¹⁴. It has been conjectured that there are infinitely many Wall-Sun-Sun primes, but the conjecture remains unproven.

Also see

• Wieferich prime
• Wilson prime
• Wolstenholme prime

External links

• The Prime Glossary: Wall-Sun-Sun prime
• Status of the search for Wall-Sun-Sun primes