Wieferich prime

In mathematics, a Wieferich prime is prime number p such that 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.

The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 (sequence A001220 in the 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.

Properties of Wieferich primes

• Wieferich primes and Mersenne numbers.

A Mersenne number is defined as M_q = 2^q −1 (where q is prime) and by Fermat's little theorem it is known, that M_p−1 (= 2^p−1−1) is always divisible by a prime p.

Moreover, it may be, that with q being a primefactor of p−1 even M_q < M_p−1 is divisible by p.

From the definition of a Wieferich prime w it is, that 2^w−1 −1 is divisible by w² and not only by w.

Now q may be a factor of w−1, and M_q still divisible by w; so the question arises, whether there exist a Mersenne number M_q, which is also divisible by w² or even may itself be a wieferich prime.

It can be shown, that

if w² divides 2^w−1−1, and w would divide M_q (= 2^q−1), where q is a primedivisor of w−1

then also w² must divide M_q; thus M_q would contain a square (and could not be prime).

The two known Wieferich primes w=1093 and w=3511 do not satisfy the condition of dividing a Mersenne number M_q with prime exponent q; so

no known Wieferich prime is a factor of a Mersenne number.

But whether this is generally impossible is not known currently; a more general notion of this question is: are all Mersenne numbers squarefree?

Since any M_q containing a Wieferich prime w must also contain w², it follows immediately, it would not be prime. Thus

a Mersenne prime cannot be a Wieferich prime.

• 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).

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}. Prime numbers of this kind have been called Mirimanoff primes on occasion, but the name has not entered general mathematical use.

See also

• Wilson prime
• Wall-Sun-Sun prime
• Wolstenholme prime

External links

• The Prime Glossary: Wieferich prime
• MathWorld: Wieferich prime
• Status of the search for Wieferich primes

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