Wieferich prime

In mathematics, a Wieferich prime is a certain kind of prime number. A prime p is called a Wieferich prime iff p² divides 2^{p − 1} − 1; compare this with Fermat's little theorem, which states that every prime p divides 2^{p − 1} − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem.

Table of contents

1 The search for Wieferich primes
2 Properties of Wieferich primes
3 Wieferich primes and Fermat's last theorem
4 Also see
5 External links
6 Further reading

The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 (Sloane's A001220), found by W. Meissner in 1913 and N. G. W. H. Beeger in 1922, respectively; if any others exist, they must be > 1.25 · 10¹⁵ [1]. It has been conjectured that only finitely many Wieferich primes exist; the conjecture remains unproven until today, although J. H. Silverman was able to show in 1988 that if the abc Conjecture holds, then for any positive integer a > 1, there exist infinitely many primes p such that p² does not divide a^{p − 1} − 1.

Properties of Wieferich primes

It can be shown that a prime factor p of a Mersenne number M_q = 2q − 1 is a Wieferich prime iff 2q − 1 divides p²; from this, it follows immediately that a Mersenne prime cannot be a Wieferich prime. Also, if p is a Wieferich prime, then 2^p² = 2 mod p².

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 natural numbers 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 p must also divide 3^{p − 1}. Prime numbers of this kind have been called Mirimanoff primes on occasion, but the name has not entered general mathematical use.