# Wall-Sun-Sun prime

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Wall-Sun-Sun prime

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" &gt; 5 is called a Wall-Sun-Sun prime if "p"² divides

:$Fleft\left(p - left\left(frac$p5 ight) ight)

where "F"("n") is the "n"th Fibonacci number and $left\left(frac$ab ight) 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;Update after|2008|12|31 if any exist, they must be &gt; 1014. It has been conjectured that there are infinitely many Wall-Sun-Sun primes.

* Wieferich prime
* Wilson prime
* Wolstenholme prime

References

*

* Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=WallSunSunPrime The Prime Glossary: Wall-Sun-Sun prime] at the Prime Pages.
*
* Richard McIntosh, [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wall-Sun-Sun primes (October 2003)]

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