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Wall-Sun-Sun prime

In mathematics, a Wall-Sun-Sun prime is a certain kind of prime number which is conjectured to exist although none are known. A prime p > 5 is called a Wall-Sun-Sun prime if p² divides

where F(n) is the nth Fibonacci number and 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 centuries-old conjecture.

No Wall-Sun-Sun primes are known as of 2008; if any exist, they must be > 1014. It has been conjectured that there are infinitely many Wall-Sun-Sun primes.

See also

* Wieferich prime

* Wilson prime

* Wolstenholme prime


* Chris Caldwell, The Prime Glossary: Wall-Sun-Sun prime at the Prime Pages.

* Eric W. Weisstein, Wall-Sun-Sun prime at MathWorld.

* Richard McIntosh, Status of the search for Wall-Sun-Sun primes (October 2003)

Number Theory

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