Wieferich prime

In number theory, 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.

The search for Wieferich primes

The only known Wieferich primes are 1093 and 3511 OEIS|id=A001220, found by W. Meissner in 1913 and N. G. W. H. Beeger in 1922, respectively; if any others exist, they must be > 1.25e|15 [http://web.archive.org/web/20060219041807/torch.cs.dal.ca/~knauer/wieferich/] . It has been conjectured that only finitely many Wieferich primes exist; the conjecture remains unproven.

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". Since Mersenne numbers of prime indices "M"_"p" and "M"_"q" are co-prime, ::A prime divisor "p" of "M"_"q", where "q" is prime, is a Wieferich prime if and only if "p"² divides "M"_"q". [http://primes.utm.edu/mersenne/index.html#unknown] :Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index 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.

* 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.

Generalizations

A Wieferich pair is a pair of primes "p" and "q" that satisfy

:"p"^{"q" − 1} ≡ 1 (mod "q"²) and "q"^{"p" − 1} ≡ 1 (mod "p"²)

so that a Wieferich prime "p" which is ≡ 1 (mod 4) will form such a pair ("p", 2): the only known instance in this case is "p" = 1093. There are 6 known Wieferich pairs.

For a cyclotomic generalisation of the Wieferich property ("n"^"p" − 1)/("n" − 1) divisible by "w"² there are solutions like :(3⁵ − 1 )/(3 − 1) = 11²and even higher exponents than 2 like in:(19⁶ − 1 )/(19 − 1) divisible by 7³.

See also

* Wieferich pair
* Wieferich@Home
* Wilson prime
* Wall-Sun-Sun prime
* Wolstenholme prime
* Taro Morishima
* Double Mersenne number

References

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*cite book | author=Richard K. Guy | authorlink = Richard K. Guy | title = Unsolved Problems in Number Theory | edition = 3rd ed | publisher = Springer Verlag | year = 2004 | isbn = 0-387-20860-7 | page = 14 .

External links

* [http://primes.utm.edu/glossary/page.php?sort=WieferichPrime The Prime Glossary: Wieferich prime]
* [http://mathworld.wolfram.com/WieferichPrime.html MathWorld: Wieferich prime]
* [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wieferich primes]