Wieferich prime
In mathematics, a Wieferich prime is prime number p such that 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.
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
External links
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 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
Referenced By
List of mathematical topics (V-Z) | List of number theory topics | Prime number | Prime numbers | Primes | Wall-Sun-Sun prime | Wilson prime | Wolstenholme prime
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