A Wall-Sun-Sun prime is a prime number $p > 5$ such that $p^2 | F_{p - \left(\frac{p}{5}\right)}$ , with $F_n$ being the $n$ th Fibonacci number and $\left(\frac{p}{5}\right)$ being a Legendre symbol. The prime $p$ always divides $F_{p - \left(\frac{p}{5}\right)}$ , but no case is known for the square of a prime $p^2$ also dividing that.

The search for these primes started in the 1990s as Donald Dines Wall, Zhi-Hong Sun and Zhi-Wei Sun searched for counterexamples to Fermat's last theorem. But Andrew Wiles's proof does not rule out the existence of these primes: if Fermat's last theorem was false and there existed a prime exponent $p$ such that $x^p + y^p = z^p$ , the square of such a prime would also divide $F_{p - \left(\frac{p}{5}\right)}$ , but with Fermat's last theorem being true, the existence of a Wall-Sun-Sun prime would not present a contradiction.

As of 2005, the lower bound was $3.2 \times 10^{12}$ , given by McIntosh.

Bibliography

1

Richard Crandall & Carl Pomerance, Prime Numbers: A Computational Perspective, 2nd Edition. New York: Springer (2005): 32