Talk:Wieferich prime

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Contents 1 Wieferich primes and Fermat's last theorem 2 Fermat numbers 2.1 Added section 2.2 Commentary

[edit] Wieferich primes and Fermat's last theorem

The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:

Let p be prime, and let x, y, z be integers such that x^p + y^p + z^p = 0. Furthermore, assume that p does not divide the product xyz. Then p is a Wieferich prime.

In 1910, Mirimanoff was able to expand the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must divide m^{p − 1}-1. He proved for m=3. Taro Morishima proved in 1931 for every prime number m not exceeding 31.

[edit] Fermat numbers

[edit] Added section

Wieferich primes and Fermat numbers.

A Fermat prime cannot also be a Wieferich prime.

• W.Banks,F. Luca and I. Shparlinski, “Estimates for Wieferich numbers,” Ramanujan J. 14. (2007), no. 3, 361–378.

[edit] Commentary

This line has been added frequently, with the reference added to the "further reading" section. This is

1. Badly written
2. Not written with a < ref > tag or {{cite}} template.
3. Not notable.
4. Sourced, if at all, to that reference, which may, in turn, not be reliable, notable, or contain the result.

— Arthur Rubin (talk) 19:28, 26 April 2008 (UTC)