Wall-Sun-Sun Prime

Let F_n be the nth Fibonacci number, and let (p|5) be a Legendre symbol so that

 e_p=(p/5)={1   for p=1,4 (mod 5); -1   for p=2,3 (mod 5).

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

 F_(p-e_p)=0 (mod p^2).

The first few values of F_(p-e_p) (mod p^2) are 2, 3, 5, 21, 55, 39, 272, 57, ... (OEIS A113650).

There are no Wall-Sun-Sun primes up to a limit of 10^(14) (McIntosh 2004), subsequently extended to 1.4597479×10^(17) by PrimeGrid as of November 2015.

Let L_n be the nth Lucas number. Then

 L_(p-e_p)=2e_p (mod p^2)

for all primes p>5. The first few values of 2e_p (mod p^2) are 2, 7, 0, 47, 2, 167, 287, ... (OEIS A113651).

Let p be prime. Then the following are equivalent:

1. F_(p-e_p)=0 (mod p^2),

2. F_p=e_p (mod p^2),

3. L_p=1 (mod p^2).

See also

Fibonacci Number, Integer Sequence Primes, Lucas Number

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McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. PRPNet. "Wall-Sun-Sun Prime Search.", N. J. A. Sequences A113650 and A113651 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Wall-Sun-Sun Prime

Cite this as:

Weisstein, Eric W. "Wall-Sun-Sun Prime." From MathWorld--A Wolfram Web Resource.

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