Eisenstein prime

For the unrelated concept of an Eisenstein prime of a modular curve, see Eisenstein ideal.

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

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

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 Gaussian primes 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, the largest known (real) Eisenstein prime is the ninth largest known prime 10223 × 2^31172165 + 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.

See also

• Gaussian prime