Wieferich prime

A Wieferich primeMathworldPlanetmath a is prime numberMathworldPlanetmath p such that p2 divides 2p-1-1; compare this with Fermat’s little theorem, which states that every prime p divides 2p-1-1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat’s last theorem.

The only known Wieferich primes are 1093 and 3511, found by W. Meissner in 1913 and N. G. W. H. Beeger in 1922, respectively; if any others exist, they must be at least 1.25×1015. The conjecture that only finitely many Wieferich primes exist remains unproven, though J. H. Silverman was able to show in 1988 that if the abc ConjectureMathworldPlanetmath holds, then for any positive integer a>1, there exist infinitely many primes p such that p2 does not divide ap-1-1. In particular, there are infinitely many primes which are not Wieferich.

Wieferich primes and Fermat’s last theorem

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

Theorem 1.

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

In 1910, Mirimanoff was able to expand this theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p-1. Prime numbers of this kind have been called Mirimanoff primes on occasion, but the name has not entered general mathematical use.

An analysis of Wieferich primes also proved crucial to Preda Mihailescu’s proof of the (formerly-named) Catalan’s conjecture.


  • 1 Ireland, Kenneth and Rosen, Michael. A Classical Introduction to Modern Number TheoryMathworldPlanetmathPlanetmath. Springer, 1998.
  • 2 Nathanson, Melvyn B. Elementary Methods in Number Theory. Springer, 2000.
  • 3 Wikipedia, the free encyclopedia, entry on Wieferich primes. All text is available under the terms of the GNU Free Documentation License
Title Wieferich prime
Canonical name WieferichPrime
Date of creation 2013-03-22 13:50:21
Last modified on 2013-03-22 13:50:21
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 11A07