Eisenstein prime

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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.

In mathematics, an Eisenstein prime is an Eisenstein integer

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

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

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

1. z is equal to the product of a unit and a natural prime of the form 3n − 1,
2. |z|² = a² − ab + b² is a natural prime (necessarily congruent to 0 or 1 modulo 3).

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

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 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.

As of 2007, the largest known (real) Eisenstein prime is 27653·2^9167433 + 1, which is the seventh largest known prime, discovered by Gordon [1]. All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes (except the smallest, 3) are congruent to 1 mod 3. Compare the Gaussian primes.