WIEFERICH PRIME

In mathematics, a 'Wieferich prime' is a prime number ''p'' such that ''p''² 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.

Contents

The search for Wieferich primes

Properties of Wieferich primes

Wieferich primes and Fermat's last theorem

See also

External links

Further reading

The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 , 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, 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 ''p''² does ''not'' divide ''a''^{''p'' − 1} − 1.

Properties of Wieferich primes

★ Wieferich primes and Mersenne numbers.
:Given a positive integer ''n'', the ''n''th Mersenne number is defined as ''M''_''n'' = 2^''n'' −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 ''p''² 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 ''p''² 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 ''w''² 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^''w''² = 2 (mod ''w''²).

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 ''p''² must also divide 3^{''p'' − 1} − 1.

See also

★ Wieferich pair

★ Wilson prime

★ Wall-Sun-Sun prime

★ Wolstenholme prime

★ Taro Morishima

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

★ T. Morishima, "Uber die Fermatsche Vermutung. XI", (German). Jap. J. Math. 11, 241-252 (1935).