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

单词	WallSunSunPrime
释义	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
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