Wall-Sun-Sun prime

If p is a prime greater than five, then p divides u(p-(p|5)), where u(n) is the nth 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 p² 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.

References:

CDP97
R. Crandall, K. Dilcher and C. 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 of a^p-1 ≡ 1 (mod p²) ," Math. Comp., 61 (1991) 361-363. MR 94d:11003
SS92
Z. Sun and Z. Sun, "Fibonacci numbers and Fermat's last theorem," Acta. Arith., 60 (1992) 371-388. MR 93e:11025
Wall60
D. D. Wall, "Fibonacci series modulo m," 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