# Wall-Sun-Sun prime

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.

## References

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

Title | Wall-Sun-Sun prime |
---|---|

Canonical name | WallSunSunPrime |

Date of creation | 2013-03-22 18:04:18 |

Last modified on | 2013-03-22 18:04:18 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A41 |

Synonym | Fibonacci Wieferich prime |