Eisenstein prime


In mathematics, an Eisenstein prime is an Eisenstein integer
that is irreducible in the ring-theoretic sense: its only Eisenstein divisors are the units, itself and its associates.
The associates and the complex conjugate of any Eisenstein prime are also prime.

Characterization

An Eisenstein integer is an Eisenstein prime if and only if either of the following conditions hold:
  1. is equal to the product of a unit and a natural prime of the form ,
  2. is a natural prime.
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, the natural Eisenstein primes are exactly the natural primes ending with 5 or . The natural Gaussian primes are exactly the natural primes ending with 7 or .

Examples

The first few Eisenstein primes that equal a natural prime are:
Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z. For example:
In general, if a natural prime p is 1 modulo 3 and can therefore be written as, then it factorizes over Z as
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.

Large primes

, the largest known Eisenstein prime is the ninth largest known prime, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to, and all Mersenne primes greater than 3 are congruent to ; thus no Mersenne prime is an Eisenstein prime.