Mersenne prime: Difference between revisions

Revision as of 00:03, 22 April 2009

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 $M_{11}=2047=23\cdot 89$ ).

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>).
+==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.