Prime number of Eisenstein

In Mathematical, a prime number of Eisenstein is a Entier of Eisenstein

$a \, \ Omega + B \,$

who is an irreducible (or in an equivalent way first) in the direction of the theory of the rings: its only dividing of Eisenstein is the units: { $1$ , $1+ \ Omega \,$ , $\ Omega \,$ , $-1$ , $-1- \ Omega \,$ , $- \ Omega \,$ }, and ' $a \, \ Omega + B \,$ itself and its units multiple. Here, $\ Omega \,$ is the Racine of the complex unit cubic

$\ frac {- 1 + I \ sqrt {3}} {2}$

The prime numbers of Eisenstein are precisely the entireties of Eisenstein $\ alpha \,$ which fill one of the following conditions:

α is equal to the product of a unit and of $1- \ Omega \,$ ,
α is of standard a prime number adequate with a modulo three,
α is the product of a unit and a natural entirety first adequate with two modulos three.

The first prime numbers of Eisenstein which are equal to a prime number naturalness 3 N - 1 are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101

who are listed in. Certain prime numbers of Eisenstein are

$2+ \ Omega \,$ , $3+ \ Omega \,$ , $4+ \ Omega \,$ , $5+2 \, \ Omega \,$ , $6+ \ Omega \,$ , $7+ \ Omega \,$ , $7+3 \, \ Omega \,$

The complex combined of any prime number of Eisenstein is another prime number of Eisenstein; by multiplying a prime number of Eisenstein by any of its units also a prime number of Eisenstein gives. The prime numbers listed above, together with 2 and 5, are all of the prime numbers of Eisenstein of absolute Value not exceeding 7.

The prime numbers of Eisenstein were named in the honor of the Mathématicien Ferdinand Eisenstein.

In 2005, the greatest prime number of Eisenstein (real) known is $27653.2 {9167433} + 1 \,$ , which is the fifth greater prime number known, discovered by Gordon. All the larger prime numbers are prime numbers of Mersenne, discovered by GIMPS. The prime numbers of Eisenstein are adequate to 2 MOD 3, and the prime numbers of Mersenne (except smallest, 3) are adequate to 1 MOD 3.

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