Let  be the th Fibonacci number, and let  be a Legendre symbol so that

{1 for p=1,4 (mod 5); -1 for p=2,3 (mod 5). " src="https://mathworld.wolfram.com/images/equations/Wall-Sun-SunPrime/NumberedEquation1.gif" />

(1)

A prime  is called a Wall-Sun-Sun prime if

(2)

The first few values of  are 2, 3, 5, 21, 55, 39, 272, 57, ... (OEIS A113650).

There are no Wall-Sun-Sun primes up to a limit of  (McIntosh 2004), subsequently extended to  by PrimeGrid as of November 2015.

Let  be the th Lucas number. Then

(3)

for all primes . The first few values of  are 2, 7, 0, 47, 2, 167, 287, ... (OEIS A113651).

Let  be prime. Then the following are equivalent:

1. ,

2. ,

3. .

REFERENCES:

McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. https://www.loria.fr/~zimmerma/records/Wieferich.status.

PrimeGrid PRPNet. "Wall-Sun-Sun Prime Search." https://prpnet.primegrid.com:13001.

Sloane, N. J. A. Sequences A113650 and A113651 in "The On-Line Encyclopedia of Integer Sequences."