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.

Eisenstein primes in a larger range

In mathematics, an Eisenstein prime is an Eisenstein integer

${\displaystyle z=a+b\,\omega ,\quad {\text{where))\quad \omega =e^((\frac {2\pi }{3))i},}$

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.

Characterization

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

1. z is equal to the product of a unit and a natural prime of the form 3n − 1 (necessarily congruent to 2 mod 3),
2. |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 primes that are prime in the Gaussian integers are exactly the natural primes ending with 7 or B, i.e., the natural primes congruent to 3 mod 4).)

Examples

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 the 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).

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.

Large primes

As of September 2019^[update], the largest known (real) Eisenstein prime is the ninth largest known prime 10223 × 2^31172165 + 1, discovered by Péter Szabolcs and PrimeGrid.^[1] 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.