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In
mathematics, the **Eisenstein integers** (named after
Gotthold Eisenstein), occasionally also known^{
[1]} as **Eulerian integers** (after
Leonhard Euler), are the
complex numbers of the form

where *a* and *b* are
integers and

is a primitive (hence non-real) cube root of unity.

The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.

The Eisenstein integers form a
commutative ring of
algebraic integers in the
algebraic number field **Q**(*ω*) – the third
cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each *z* = *a* + *bω* is a root of the
monic polynomial

In particular, *ω* satisfies the equation

The product of two Eisenstein integers *a* + *bω* and *c* + *dω* is given explicitly by

The 2-norm of an Eisenstein integer is just its squared modulus, and is given by

which is clearly a positive ordinary (rational) integer.

Also, the
complex conjugate of *ω* satisfies

The
group of units in this ring is the
cyclic group formed by the sixth
roots of unity in the complex plane: {±1, ±*ω*, ±*ω*^{2}}, the Eisenstein integers of norm 1.

The ring of Eisenstein integers forms a
Euclidean domain whose norm *N* is given by the square modulus, as above:

A
division algorithm, applied to any dividend *α* and divisor *β* ≠ 0, gives a quotient *κ* and a remainder *ρ* smaller than the divisor, satisfying:

Here, *α*, *β*, *κ*, *ρ* are all Eisenstein integers. This algorithm implies the
Euclidean algorithm, which proves
Euclid's lemma and the
unique factorization of Eisenstein integers into Eisenstein primes.

One division algorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms of *ω*:

for rational *a*, *b* ∈ **Q**. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:

Here may denote any of the standard rounding-to-integer functions.

The reason this satisfies *N*(*ρ*) < *N*(*β*), while the analogous procedure fails for most other
quadratic integer rings, is as follows. A fundamental domain for the ideal **Z***ω**β* = **Z***β* + **Z***ωβ*, acting by translations on the complex plane, is the 60°–120° rhombus with vertices 0, *β*, *ωβ*, *β* + *ωβ*. Any Eisenstein integer *α* lies inside one of the translates of this parallelogram, and the quotient *κ* is one of its vertices. The remainder is the square distance from *α* to this vertex, but the maximum possible distance in our algorithm is only , so . (The size of *ρ* could be slightly decreased by taking *κ* to be the closest corner.)

If *x* and *y* are Eisenstein integers, we say that *x* divides *y* if there is some Eisenstein integer *z* such that *y* = *zx*. A non-unit Eisenstein integer *x* is said to be an Eisenstein prime if its only non-unit divisors are of the form *ux*, where *u* is any of the six units. They are the corresponding concept to the
Gaussian primes in the Gaussian integers.

There are two types of Eisenstein prime. First, an ordinary
prime number (or *rational prime*) which is congruent to 2 mod 3 is also an Eisenstein prime. Second, 3 and each rational prime congruent to 1 mod 3 are equal to the norm *x*^{2} − *xy* + *y*^{2} of an Eisentein integer *x* + *ωy*. Thus, such a prime may be factored as (*x* + *ωy*)(*x* + *ω*^{2}*y*), and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.

The first few Eisenstein primes of the form 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*).

In general, if a natural prime *p* is 1 modulo 3 and can therefore be written as *p* = *a*^{2} − *ab* + *b*^{2}, then it factorizes over **Z***ω* as

- p = (
*a*+*bω*)((*a*−*b*) −*bω*).

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.

As of October 2023^{
[update]}, the largest known real Eisenstein prime is the
tenth-largest known prime 10223 × 2^{31172165} + 1, discovered by Péter Szabolcs and
PrimeGrid.^{
[2]} With one exception,^{[
clarification needed]} all larger known primes are
Mersenne primes, discovered by
GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and all Mersenne primes greater than 3 are congruent to 1 mod 3; thus no Mersenne prime is an Eisenstein prime.

The sum of the reciprocals of all Eisenstein integers excluding 0 raised to the sixth power can be expressed in terms of the gamma function:

where

The
quotient of the complex plane **C** by the
lattice containing all Eisenstein integers is a
complex torus of real dimension 2. This is one of two tori with maximal
symmetry among all such complex tori.^{[
citation needed]} This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. (The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of
Gaussian integers, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as [0, 1] × [0, 1].)

- Gaussian integer
- Cyclotomic field
- Systolic geometry
- Hermite constant
- Cubic reciprocity
- Loewner's torus inequality
- Hurwitz quaternion
- Quadratic integer
- Dixon elliptic functions

**^**Both Surányi, László (1997).*Algebra*. TYPOTEX. p. 73. and Szalay, Mihály (1991).*Számelmélet*. Tankönyvkiadó. p. 75. call these numbers "Euler-egészek", that is, Eulerian integers. The latter claims Euler worked with them in a proof.**^**"Largest Known Primes".*The Prime Pages*. Retrieved 2023-02-27.**^**"Entry 0fda1b – Fungrim: The Mathematical Functions Grimoire".*fungrim.org*. Retrieved 2023-06-22.