Wieferich prime

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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.

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

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The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 Template: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.

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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).

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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.

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External links

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Wieferich prime

From Free net encyclopedia

Contents

The search for Wieferich primes

Properties of Wieferich primes

Wieferich primes and Fermat's last theorem

See also

External links

Further reading

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