Eisenstein prime

In mathematics, an Eisenstein prime is an Eisenstein integer

: a ω + b

that is

:frac{-1 + isqrt{3}}{2}

Cox and Wagon proved that besides 1 - ω, there are only three kinds of Eisenstein primes:

# a ω + b such that a^2 - ab + b^2 is a natural prime number of the form 3n + 1. # a ω2 + b such that a^2 - ab + b^2 is a natural prime number of the form 3n + 1. # a ω + b where a = 0 (and thus there is no imaginary part) and b is a natural prime number of the form 3n - 1.

Therefore, this last kind of Eisenstein prime is also a kind of natural prime. In fact, all natural primes of the form 3 n -1 are Eisenstein primes. The first few Eisenstein primes of this form are:

2 (number), 5 (number), 11 (number), 17 (number), 23 (number), 29 (number), 41 (number), 47 (number), 53 (number), 59 (number), 71 (number), 83 (number), 89 (number), 101 (number)

which are listed in . Some non-real Eisenstein primes are

2+ω, 3+ω, 4+ω, 5+2ω, 6+ω, 7+ω, 7+3ω

The complex conjugate of any Eisenstein prime is another; multiplying an Eisenstein prime by any of the units 1, 1+ω, ω, -1, -1-ω, -ω also gives an Eisenstein prime. 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.

. Real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes (except the smallest, 3) are congruent to 1 mod 3.