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 , itself and its associates. The associates (unit multiples) 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 (mutually exclusive) conditions hold: 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, , ): The natural Eisenstein primes are exactly the natural primes ending with 5 or (i.e. the natural primes congruent to ). (The natural primes that are prime in the Gaussian integers are exactly the natural primes ending with 7 or , i.e., the natural primes congruent to ).) 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["ω"]. For example: In general, if a natural prime "p" is 1 modulo 3 and can  therefore be written as , then it factorizes over Z["ω"] 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 (real) 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.