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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:
- z is equal to the product of a unit and a natural prime of the form 3n − 1 (necessarily congruent to 2 mod 3),
- |z|2 = a2 − ab + 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 (written with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B), the natural Eisenstein primes are exactly the natural primes ending with 5 or B (i.e. the natural primes congruent to 2 mod 3). The natural Gaussian primes are exactly the natural primes ending with 7 or B (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:
- 3 = −(1 + 2ω)2
- 7 = (3 + ω)(2 − ω).
In general, if a natural prime p is 1 modulo 3 and can therefore be written as p = a2 − ab + b2, then it factorizes over Z[ω] as
- p = (a + bω)((a − b) − bω).
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 September 2019[update], the largest known (real) Eisenstein prime is the ninth 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 all Mersenne primes greater than 3 are congruent to 1 mod 3; thus no Mersenne prime is an Eisenstein prime.