Eisenstein prime

In mathematics, an Eisenstein prime is an Eisenstein integer

:$z\; =\; a\; +\; b,omegaqquad(omega\; =\; e^\{2pi\; i/3\})$

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

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

An Eisenstein integer "z" = "a" + "b"ω is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions holds:
#"z" is equal to the product of a unit and a natural prime of the form 3"n" − 1,
#|"z"|² = "a"² − "ab" + "b"² 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 3"n" − 1 are:

:2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101 OEIS|id=A003627

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 [http://primes.utm.edu/top20/page.php?id=3] . 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.