Wall-Sun-Sun prime

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In number theory, a Wall-Sun-Sun prime is a certain kind of prime number which is conjectured to exist although none are known. A prime p > 5 is called a Wall-Sun-Sun prime if p² divides

$F\left(p - \left(\frac{{p}}{{5}}\right)\right)$

where F(n) is the nth Fibonacci number and $\left(\frac{{a}}{{b}}\right)$ is the Legendre symbol of a and b.

Wall-Sun-Sun primes are named after D. D. Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime p, then p would have to be a Wall-Sun-Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall-Sun-Sun primes was also the search for a counterexample to this centuries-old conjecture.

No Wall-Sun-Sun primes are known as of 2008; if any exist, they must be > 10¹⁴. It has been conjectured that there are infinitely many Wall-Sun-Sun primes.