Eisenstein prime

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In mathematics, an Eisenstein prime is an Eisenstein integer

aω + b

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units 1, 1+ω, ω, -1, -1-ω, -ω, and aω + b itself and its unit multiples. Here ω is the complex cube root of unity

\frac{-1 + i\sqrt{3}}{2}

The Eisenstein primes are precisely those Eisenstein integers α which fulfil one of the following conditions:

  1. α is equal to the product of a unit and 1 - ω,
  2. α is equal to the product of a unit and a natural prime 3n - 1,
  3. α can be multiplied by an Eisenstein integer such that the product is a natural prime 3n + 1.

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

which are listed in (sequence A003627 in OEIS). Some non-real Eisenstein primes are

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

The complex conjugate of any Eisenstein prime is another; multiplying an Eisenstein prime by any of the units 1, 1+ω, ω, -1, -1-ω, -ω also gives an Eisenstein prime. 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.

Eisenstein primes are named after the mathematician Ferdinand Eisenstein.

As of 2005, the largest known (real) Eisenstein prime is 27653·29167433 + 1, which is the fifth 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.

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