Eisenstein prime

Given the complex cubic root of unity $\omega =e^{\frac{2i\pi }{3}}$, 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 $p\equiv 2mod3$. Therefore no Mersenne prime is also an Eisenstein prime.