Wieferich prime
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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.
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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] (http://www.cs.dal.ca/~knauer/wieferich/). 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 integers 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
- The Prime Glossary: Wieferich prime (http://primes.utm.edu/glossary/page.php?sort=WieferichPrime)
- MathWorld: Wieferich prime (http://mathworld.wolfram.com/WieferichPrime.html)
- Status of the search for Wieferich primes (http://www.loria.fr/~zimmerma/records/Wieferich.status)
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) 226-237de:Wieferich-Primzahl