# Wall-Sun-Sun prime

If*p*is a prime greater than five, then

*p*divides

*u*(

*p*-(

*p*|5)), where

*u*(

*n*) is the

*n*th Fibonacci number and (

*a*|

*b*) is the Legendre symbol (so (

*p*|5)=1 when

*p*is a multiple of 5 plus either 1 or 4, and (

*p*|5)=-1 when

*p*is a multiple of 5 plus either 2 or 3). A prime

*p*> 5 is a

**Wall-Sun-Sun prime**if

*p*

^{2}divides

*u*(

*p*-(

*p*|5)). These are sometimes called Fibonacci-Wieferich primes.

*No Wall-Sun-Sun primes are known*
(and mathematicans have checked all primes
below 100,000,000,000,000)!
So why bother naming a type of prime when we know
of no such numbers? For the following two reasons.

First Sun and Sun
showed in 1992 that if the first case of Fermat's
Last Theorem (FLT) was false for the prime exponent
*p*, then *p* is a Wall-Sun-Sun prime.
Before Wiles proved FLT, the search for Wall-Sun-Sun
primes was also the search for a counter-example to
this theorem. Notice that this is the same reason that
Sophie Germain primes first sparked mathematicians'
interest.

Second, heuristically it seems
likely that there would be infinitely many such primes,
but that they should be very rare (just as
is conjectured for the Wilson primes and Wieferich
primes). But we must admit this heuristic is based
on the assumption that *u*(*p*-(*p*|5))/*p* behaves essentially
randomly modulo *p*, and this assumption is
made simply because we do not yet know otherwise.

**See Also:** WilsonPrime, WieferichPrime, SophieGermainPrime

**References:**

- CDP97
R. Crandall,K. DilcherandC. Pomerance, "A search for Wieferich and Wilson primes,"Math. Comp.,66:217 (1997) 433--449.MR 97c:11004(Abstract available)- Montgomery91
P. Montgomery, "New solutions ofa^{p-1}≡ 1 (mod p^{2}) ,"Math. Comp.,61(1991) 361-363.MR 94d:11003- SS92
Z. SunandZ. Sun, "Fibonacci numbers and Fermat's last theorem,"Acta. Arith.,60(1992) 371-388.MR 93e:11025- Wall60
D. D. Wall, "Fibonacci series modulom,"Amer. Math. Monthly,67(1960) 67.MR 22:10945- Williams82
H. C. Williams, "The influence of computers in the development of number theory,"Comput. Math. with Appl.,8(1982) 75-93.MR 83c:10002