Eisenstein prime

Given the complex cubic root of unity $\omega=e^{{2i\pi}\over{3}}$ , an Eisenstein integer $a\omega+b$ (where $a$ and $b$ are natural integers) is said to be an Eisenstein prime if its only divisors are 1, $\omega$ , $1+\omega$ and itself.

Eisenstein primes of the form $0\omega+b$ are ordinary natural primes $p\equiv 2\mod 3$ . Therefore no Mersenne prime is also an Eisenstein prime.