## You are here

HomeEisenstein prime

## Primary tabs

# Eisenstein prime

Given the complex cubic root of unity $\omega=e^{{{2i\pi}\over{3}}}$, an Eisenstein integer $a\omega+b$ (where $a$ and $b$ are natural integers) is said to be an *Eisenstein prime* if its only divisors are 1, $\omega$, $1+\omega$ and itself.

Eisenstein primes of the form $0\omega+b$ are ordinary natural primes $p\equiv 2\mod 3$. Therefore no Mersenne prime is also an Eisenstein prime.

Related:

EisensteinIntegers

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11R04*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Eisenstein factorization of other primes

So what's the Eisenstein integer factorization of an integer that is prime in regular integers but not congruent to 2 mod 3?

## Re: Eisenstein factorization of other primes

(Guy 2004) p. 56 gives six examples (forgive my temporary overloading of lowercase w for lowercase omega):

3 = (1 - w)(1 - w^2)

7 = (2 - w)(2 - w^2)

13 = (3 - w)(3 - w^2)

19 = (3 - 2w)(3 - 2w^2)

etc.