Wall-Sun-Sun prime

A Wall-Sun-Sun primeMathworldPlanetmath is a prime numberMathworldPlanetmath p>5 such that p2|Fp-(p5), with Fn being the nth Fibonacci numberMathworldPlanetmath and (p5) being a Legendre symbolMathworldPlanetmath. The prime p always divides Fp-(p5), but no case is known for the square of a prime p2 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 theoremMathworldPlanetmath. 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 xp+yp=zp, the square of such a prime would also divide Fp-(p5), but with Fermat’s last theorem being true, the existence of a Wall-Sun-Sun prime would not present a contradictionMathworldPlanetmathPlanetmath.

As of 2005, the lower bound was 3.2×1012, given by McIntosh.


  • 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