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.

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