Wall–Sun–Sun prime

Unsolved problems in mathematics

Are there any Wall–Sun–Sun primes?

In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist although none are known.

Definition

A prime p > 5 is called a Wall–Sun–Sun prime if p² divides the Fibonacci number $F_{p - \left(\frac{{p}}{{5}}\right)}$, where the Legendre symbol $\left(\frac{{p}}{{5}}\right)$ is defined as

$\left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5\\ -1 &\text{if }p \equiv \pm2 \pmod 5 \end{cases}$

Existence

It has been conjectured that there are infinitely many Wall–Sun–Sun primes. No Wall–Sun–Sun primes are known as of October 2011.

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2×10¹⁴. Dorais and Klyve extended this range to 9.7×10¹⁴ without finding such a prime.

History

Wall–Sun–Sun primes are named after D. D. Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a counterexample to this centuries-old conjecture.

See also

• Wieferich prime
• Wilson prime
• Wolstenholme prime
• PrimeGrid

Further reading

• Crandall, Richard E.; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, Springer, p. 29, ISBN 0387947779 

External links

• Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
• Weisstein, Eric W., "Wall–Sun–Sun prime" from MathWorld.
• Richard McIntosh, Status of the search for Wall–Sun–Sun primes (October 2003)