ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# 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 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.

 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 ap − 1 − 1.

## Properties of Wieferich primes

It can be shown that a prime factor p of a Mersenne number Mq = 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 2p² = 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 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 the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p 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.

## Also see

• A. Wieferich, "Zum letzten Fermat'schen Theorem", Journal für Reine Angewandte Math., 136 (1909) 293-302
• N. G. W. H. Beeger, "On a new case of the congruence 2p − 1 = 1 (p2), Messenger of Math, 51 (1922), 149-150
• W. Meissner, "Über die Teilbarkeit von 2pp − 2 durch das Quadrat der Primzahl p=1093, Sitzungsber. Akad. d. Wiss. Berlin (1913), 663-667
• J. H. Silverman, "Wieferich's criterion and the abc-conjecture", Journal of Number Theory, 30:2 (1988) 226-237

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.