Wall–Sun–Sun prime

A Wall - Sun - Sun primes, named after DD Wall, Zhi- Hong Sun and Zhi- Wei Sun, is a prime p > 5, for which the number divisible by p

Is divisible by p2. Where F ( n) is the n-th Fibonacci number and the Legendre symbol of a and b, that is 1 if 5 is a divisor of P2-1, and -1 else DD wall 1960 made ​​whether such primes. The question is open to this day, in particular, no Wall - Sun - Sun primes known. When a Wall - Sun - Sun prime number exists, it must be greater than 9.7 × 1014. There is the assumption that infinitely many exist.

Zhi- Hong Sun and Zhi- Wei Sun showed in 1992 that an odd prime p is a Wall - Sun - Sun prime number when a specific counter-example to Fermat's theorem exists, namely not x divisible by p integers, y, z with xp yp = zp. This property also had Wieferich 1909 demonstrated for Wieferich primes. With the proof of the conjecture in 1995 but has been clarified that no counterexample exists, so the condition can not be met.