Wall-Sun-Sun prime

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

${\displaystyle F{\bigl (}p-{\bigl (}{\tfrac {p}{5}}{\bigr )}{\bigr )}}$

is divisible by . Here, F ( n ) is the n th Fibonacci number and the Legendre symbol of a and b , that is 1 if 5 divides , and otherwise. DD Wall presented in 1960, the question of whether such primes. The question is still open today, in particular no Wall-Sun-Sun prime numbers are known. If a Wall-Sun-Sun prime exists, it must be greater than 9.7 × 10 ¹⁴ . There is a presumption that there are infinitely many. ${\displaystyle p^{2}}$ ${\displaystyle {\bigl (}{\tfrac {a}{b}}{\bigr )}}$ ${\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}$ ${\displaystyle p^{2}-1}$ ${\displaystyle -1}$

Zhi-Hong Sun and Zhi-Wei Sun showed in 1992 that an odd prime p is a Wall-Sun-Sun prime if there is a certain counter-example to Fermat's conjecture , namely integers x , y , z with x that are not divisible by p ^p + y ^p = z ^p . Wieferich had also proven this property in 1909 for Wieferich prime numbers . With the proof of the conjecture in 1995, however, it is clear that no counterexample exists, i.e. that the requirement cannot be met.

See also

• Wilson prime number
• Wolstenholme prime number

Web links

• The Prime Glossary: ​​Wall-Sun-Sun prime on the Prime Pages by Chris K. Caldwell (English)
• Eric W. Weisstein : Wall-Sun-Sun prime . In: MathWorld (English).

Individual evidence

1. ↑ DD Wall : Fibonacci series modulo m . In: American Mathematical Monthly , 67, 1960, pp. 525-532 (English)
2. ↑ François G. Dorais, Dominic W. Klyve: A Wieferich prime search up to 6.7 x 10 ¹⁵ . In: Journal of Integer Sequences , October 14, 16, 2011, Article 11.9.2
3. ↑ Jiří Klaška: Short remark on Fibonacci Wieferich primes . In: Acta Mathematica Universitatis Ostraviensis , 15, 2007, pp. 21-25 (English)
4. ↑ Zhi-Hong Sun , Zhi-Wei Sun : Fibonacci numbers and Fermat's last theorem . (PDF; 186 kB) In: Acta Arithmetica , 60, 1992, pp. 371-388