Eisenstein 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 hold:
It follows that the absolute value squared 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 Ɛ (i.e. the natural primes congruent to 2 mod 3), the natural Gaussian primes are exactly the natural primes end with 7 or Ɛ (i.e. the natural primes congruent to 3 mod 4).
The first few Eisenstein primes that equal a natural prime 3n − 1 are:
Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example:
Some non-real Eisenstein primes are
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 March 2017[update], the largest known (real) Eisenstein prime is the seventh largest known prime 10223 × 231172165 + 1, discovered by Péter Szabolcs and PrimeGrid. 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; thus no Mersenne prime is an Eisenstein prime.
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