- Wall-Sun-Sun prime
In

number theory , a**Wall-Sun-Sun prime**is a certain kind ofprime number which is conjectured to exist although none are known. A prime "p" > 5 is called a Wall-Sun-Sun prime if "p"² divides:$Fleft(p\; -\; left(frac$p5 ight) ight)

where "F"("n") is the "n"th

Fibonacci number and $left(frac$ab ight) is theLegendre symbol of "a" and "b".Wall-Sun-Sun primes are named after

D. D. Wall ,Zhi Hong Sun andZhi Wei Sun ; Z. H. Sun and Z. W. Sun showed in1992 that if the first case ofFermat'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 toAndrew Wiles ' proof of Fermat's last theorem, the search for Wall-Sun-Sun primes was also the search for acounterexample to this centuries-oldconjecture .No Wall-Sun-Sun primes are known as of 2008;Update after|2008|12|31 if any exist, they must be > 10

^{14}. It has been conjectured that there are infinitely many Wall-Sun-Sun primes.**See also***

Wieferich prime

*Wilson prime

*Wolstenholme prime **References***

**External links*** Chris Caldwell, [

*http://primes.utm.edu/glossary/page.php?sort=WallSunSunPrime The Prime Glossary: Wall-Sun-Sun prime*] at thePrime Pages .

*

* Richard McIntosh, [*http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wall-Sun-Sun primes (October 2003)*]

*Wikimedia Foundation.
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### Look at other dictionaries:

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