# Wall-Sun-Sun prime

A 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.

## References

• 1 Richard Crandall & Carl Pomerance, Prime Numbers: A Computational Perspective, 2nd Edition. New York: Springer (2005): 32
Title Wall-Sun-Sun prime WallSunSunPrime 2013-03-22 18:04:18 2013-03-22 18:04:18 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A41 Fibonacci Wieferich prime