# Eisenstein prime

Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3n − 1. All others have an absolute value squared equal to a natural prime.
Eisenstein primes in a larger range.

In mathematics, an Eisenstein prime is an Eisenstein integer

${\displaystyle z{=}a+b\,\omega \qquad {\text{where}}\quad \omega {=}e^{\frac {2\pi i}{3}}}$

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units {±1, ±ω, ±ω2}, a + itself and its associates.

The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.

## Characterization

An Eisenstein integer z = a + is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:

1. z is equal to the product of a unit and a natural prime of the form 3n − 1,
2. |z|2 = a2ab + b2 is a natural prime (necessarily congruent to 0 or 1 mod 3).

It follows that the square of the absolute value of every Eisenstein prime is a natural prime or the square of a natural prime.

In base 12, the natural Eisenstein primes are exactly the natural primes end with 5 or 3 (i.e. the natural primes congruent to 2 mod 3), the natural Gaussian primes are exactly the natural primes end with 7 or 3 (i.e. the natural primes congruent to 3 mod 4).

## Examples

The first few Eisenstein primes that equal a natural prime 3n − 1 are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... (sequence A003627 in the OEIS).

Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example:

3 = −(1 + 2ω)2
7 = (3 + ω)(2 - ω).

Some non-real Eisenstein primes are

2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω.

Up to conjugacy and unit multiples, the primes listed above, together with 2 and 5, are all the Eisenstein primes of absolute value not exceeding 7.

## Large primes

As of March 2017, the largest known (real) Eisenstein prime is the seventh largest known prime 10223 × 231172165 + 1, discovered by Péter Szabolcs and PrimeGrid.[1] All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and all Mersenne primes are congruent to 0 or 1 mod 3; thus no Mersenne prime is an Eisenstein prime.