- Wieferich prime
In

number theory , a**Wieferich prime**is aprime number "p" such that "p"^{2}divides 2^{"p" − 1}− 1; compare this withFermat's little theorem , which states that every odd prime "p" divides 2^{"p" − 1}− 1. Wieferich primes were first described byArthur Wieferich in 1909 in works pertaining toFermat'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 number s.: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. ByFermat'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

integer s 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 :(3^{5}− 1 )/(3 − 1) = 11^{2}and even higher exponents than 2 like in:(19^{6}− 1 )/(19 − 1) divisible by 7^{3}.**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*]

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**Wieferich at Home**— Wieferich@Home is the Internet based distributed computing (DC) project searching for Wieferich primes. It is the first Czech DC project. The only known Wieferich primes are 1093 and 3511, found in 1913 and in 1922, respectively. It is not known … Wikipedia**Wieferich-Primzahl**— Eine Wieferich Primzahl ist eine Primzahl p mit der Eigenschaft, dass 2p−1 − 1 durch p2 teilbar ist. Alternativ kann man dies auch als Kongruenz schreiben: Solche Primzahlen wurden 1909 von dem deutschen Mathematiker Arthur Wieferich… … Deutsch Wikipedia**Wieferich pair**— In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy: p q − 1 ≡ 1 (mod q 2) and q p − 1 ≡ 1 (mod p 2)Wieferich pairs are named after German mathematician Arthur Wieferich.The only known Wieferich pairs (sequence… … Wikipedia**Prime number**— Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… … Wikipedia**Mersenne prime**— Named after Marin Mersenne Publication year 1536[1] Author of publication Regius, H. Number of known terms 47 Conjectured number of terms Infinite … Wikipedia**Wolstenholme prime**— In number theory, a Wolstenholme prime is a certain kind of prime number. A prime p is called a Wolstenholme prime iff the following condition holds::2p 1}choose{p 1 equiv 1 pmod{p^4}.Wolstenholme primes are named after Joseph Wolstenholme who… … Wikipedia**Wall-Sun-Sun prime**— In number theory, a Wall Sun Sun prime is a certain kind of prime number which is conjectured to exist although none are known. A prime p gt; 5 is called a Wall Sun Sun prime if p ² divides :Fleft(p left(fracp5 ight) ight)where F ( n ) is the n… … Wikipedia**Arthur Wieferich**— Arthur Josef Alwin Wieferich (April 27, 1884 – September 15, 1954) was a German mathematician and teacher, remembered for his work on number theory. He was born in Münster, attended the University of Münster (1903–1909) and then worked widely as… … Wikipedia**Wilson prime**— A Wilson prime is a prime number p such that p ² divides ( p − 1)! + 1, where ! denotes the factorial function; compare this with Wilson s theorem, which states that every prime p divides ( p − 1)! + 1.The only known Wilson primes are 5, 13, and… … Wikipedia**List of prime numbers**— This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. By Euclid s theorem, there are an infinite number of prime numbers. Subsets of the… … Wikipedia