Wall-Sun-Sun prime


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" > 5 is called a Wall-Sun-Sun prime if "p"² divides

:Fleft(p - left(fracp5 ight) ight)

where "F"("n") is the "n"th Fibonacci number and left(fracab 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 > 1014. 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 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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