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 {±1, ±*ω*, ±*ω*^{2}}, *a* + *bω* itself and its associates.

The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.

## Characterization

An Eisenstein integer *z* = *a* + *bω* is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:

*z*is equal to the product of a unit and a natural prime of the form 3*n*− 1,- |
*z*|^{2}=*a*^{2}−*ab*+*b*^{2}is a natural prime (necessarily congruent to 0 or 1 mod 3).

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 end with 5 or 3 (i.e. the natural primes congruent to 2 mod 3), the natural Gaussian primes are exactly the natural primes end with 7 or 3 (i.e. the natural primes congruent to 3 mod 4).

## Examples

The first few Eisenstein primes that equal a natural prime 3*n* − 1 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:

- 3 = −(1 + 2
*ω*)^{2} - 7 = (3 +
*ω*)(2 -*ω*).

Some non-real Eisenstein primes are

- 2 +
*ω*, 3 +*ω*, 4 +*ω*, 5 + 2*ω*, 6 +*ω*, 7 +*ω*, 7 + 3*ω*.

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

As of March 2017^{[update]}, the largest known (real) Eisenstein prime is the seventh largest known prime 10223 × 2^{31172165} + 1, discovered by Péter Szabolcs and PrimeGrid.^{[1]} All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and all Mersenne primes are congruent to 0 or 1 mod 3; thus no Mersenne prime is an Eisenstein prime.

## See also

## References

**^**Chris Caldwell, "The Top Twenty: Largest Known Primes" from The Prime Pages. Retrieved 2017-03-14.