A Wall-Sun-Sun prime is a prime number 
satisfying
 |
(1)
|
where 
is the 
th
Fibonacci number and 
is the Legendre symbol
 |
(2)
|
The following residue sequences range over all primes, including 2, 3, and 5.
The first few values of 
are 2, 3, 5, 21, 55, 39, 272, 57, ... (OEIS
A113650).
McIntosh (2004) reported that there are no Wall-Sun-Sun primes up to 
. This search limit was 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. 
.
