Wieferich prime

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In mathematics, a Wieferich prime is a prime number p such that p² divides 2^{p − 1} − 1; compare this with Fermat's little theorem, which states that every odd 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.

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

[edit] The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 (sequence A001220 in OEIS), 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.

[edit] Properties of Wieferich primes

Wieferich primes and Mersenne numbers.

Given a positive integer n, the nth Mersenne number is defined as M_n = 2ⁿ −1. It is known that M_n is prime only if n is prime. By Fermat's little theorem it is known that M_p−1 (= 2^p−1−1) is always divisible by a prime p. If q is an odd prime, it can be shown that

A prime divisor p of M_q is a Wieferich prime if and only if p² divides M_q.

Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers are square-free. If a Mersenne number M_q is not square-free (i.e., there exists some prime p for which p² divides M_q), then M_q has a Wieferich prime divisor. If there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers that are not square-free.

Cyclotomic generalization

For a cyclotomic generalization of the Wieferich property (n^p−1)/(n−1) divisible by w² there are solutions like

(3⁵ - 1 )/(3-1) = 11²

and even higher exponents than 2 like in

(19⁶ - 1 )/(19-1) divisible by 7³

Also, if w is a Wieferich prime, then 2^w² = 2 (mod w²).

[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 also divide 3^{p − 1} − 1.

[edit] See also

[edit] External links

[edit] Further reading

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 2^{p − 1} = 1 (p²), Messenger of Math, 51 (1922), 149-150
W. Meissner, "Über die Teilbarkeit von 2p^p − 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
T. Morishima, "Uber die Fermatsche Vermutung. XI", (German). Jap. J. Math. 11, 241-252 (1935).