Mersenne prime: Difference between revisions

Latest revision as of 19:01, 2 January 2012

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
View other properties of prime numbers | View other properties of natural numbers

Definition

A Mersenne prime is a Mersenne number that is also a prime number. In other words, it is a number of the form $M_{n}=2^{n}-1$ that is prime, where $n$ is a natural number.

It turns out that if $M_{n}$ is prime, then $n$ itself is also prime, though the converse is not true (the smallest counterexample is $n=11$ , because $M_{11}=2047=23\cdot 89$ ).

Facts

Facts in number theory

Facts in other branches of mathematics

Order is product of Mersenne prime and one more implies normal Sylow subgroup (fact about groups)

Occurrence

Initial examples

The Mersenne numbers $M_{p}$ are prime for $p=2,3,5,7$ , with the corresponding primes $M_{p}$ being $3,7,31,127$ . Them smallest prime $p$ for which the Mersenne number $M_{p}$ is not prime is $11$ : $M_{11}=2047=23\cdot 89$ .

Infinitude conjecture

Further information: Infinitude conjecture for Mersenne primes

It is conjectured that there are infinitely many Mersenne primes.

@@ Line 3: / Line 3: @@
 ==Definition==
-A '''Mersenne prime''' is a [[Mersenne number]] that is also a [[prime number]]. In other words, it is a number of the form <math>M_n = 2^n - 1</math> that is prime, where <math>n</math> is a natural number.
+A '''Mersenne prime''' is a [[defining ingredient::Mersenne number]] that is also a [[prime number]]. In other words, it is a number of the form <math>M_n = 2^n - 1</math> that is prime, where <math>n</math> is a natural number.
 It turns out that if <math>M_n</math> is prime, then <math>n</math> itself is also prime, though the converse is not true (the smallest counterexample is <math>n = 11</math>, because <math>M_{11} = 2047 = 23 \cdot 89</math>).
+==Facts==
+===Facts in number theory===
+* [[Mersenne number is prime implies number is prime]]
+* [[Mersenne number for prime or Poulet implies prime or Poulet]]
+===Facts in other branches of mathematics===
+* [[Groupprops:Order is product of Mersenne prime and one more implies normal Sylow subgroup|Order is product of Mersenne prime and one more implies normal Sylow subgroup (fact about groups)]]
+==Occurrence==
+===Initial examples===
+The Mersenne numbers <math>M_p</math> are prime for <math>p = 2,3,5,7</math>, with the corresponding primes <math>M_p</math> being <math>3,7,31,127</math>. Them smallest prime <math>p</math> for which the Mersenne number <math>M_p</math> is ''not'' prime is <math>11</math>: <math>M_{11} = 2047 = 23 \cdot 89</math>.
+===Infinitude conjecture===
+{{further|[[Infinitude conjecture for Mersenne primes]]}}
+It is conjectured that there are infinitely many Mersenne primes.