Inmathematics, theEisenstein integers (named afterGotthold Eisenstein), occasionally also known^[1] asEulerian integers (afterLeonhard Euler), are thecomplex numbers of the form

${\displaystyle z=a+b\omega ,}$

wherea andb areintegers and

${\displaystyle \omega =\frac{-1+i\sqrt{3}}{2}=e^{i2\pi /3}}$

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

The Eisenstein integers form atriangular lattice in thecomplex plane, in contrast with theGaussian integers, which form asquare lattice in the complex plane. The Eisenstein integers are acountably infinite set.

The Eisenstein integers form acommutative ring ofalgebraic integers in thealgebraic number fieldQ(ω) – the thirdcyclotomic field. To see that the Eisenstein integers are algebraic integers note that eachz =a +bω is a root of themonic polynomial

${\displaystyle z^2-(2a-b)z+\left(a^2-ab+b^2\right).}$

In particular,ω satisfies the equation

${\displaystyle {\omega }^2+\omega +1=0.}$

The product of two Eisenstein integersa +bω andc +dω is given explicitly by

${\displaystyle (a+b\omega )(c+d\omega )=(ac-bd)+(bc+ad-bd)\omega .}$

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

${\displaystyle {\left|a+b\omega \right|}^2={(a-\frac{1}{2}b)}^2+\frac{3}{4}{b}^2=a^2-ab+b^2,}$

which is clearly a positive ordinary (rational) integer.

Also, thecomplex conjugate ofω satisfies

${\displaystyle \overline{\omega }={\omega }^2.}$

Thegroup of units in this ring is thecyclic group formed by the sixthroots of unity in the complex plane:{±1, ±ω, ±ω²}, the Eisenstein integers of norm 1.

The ring of Eisenstein integers forms aEuclidean domain whose normN is given by the square modulus, as above:

${\displaystyle N(a+b\omega )=a^2-ab+b^2.}$

Adivision algorithm, applied to any dividendα and divisorβ ≠ 0, gives a quotientκ and a remainderρ smaller than the divisor, satisfying:

Here,α,β,κ,ρ are all Eisenstein integers. This algorithm implies theEuclidean algorithm, which provesEuclid's lemma and theunique 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 ofω:

${\displaystyle \frac{\alpha }{\beta }=\frac{1}{|\beta {|}^2}\alpha \overline{\beta }=a+bi=a+\frac{1}{\sqrt{3}}b+\frac{2}{\sqrt{3}}b\omega ,}$

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

${\displaystyle \kappa =\lfloor a+\frac{1}{\sqrt{3}}b\rceil +\lfloor \frac{2}{\sqrt{3}}b\rceil \omega \text{ and }\rho =\alpha -\kappa \beta .}$

Here${\displaystyle \lfloor x\rceil }$ may denote any of the standardrounding-to-integer functions.

The reason this satisfiesN(ρ) <N(β), while the analogous procedure fails for most otherquadratic integer rings, is as follows. A fundamental domain for the idealZ[ω]β =Zβ +Zωβ, acting by translations on the complex plane, is the 60°–120° rhombus with vertices0,β,ωβ,β +ωβ. Any Eisenstein integerα lies inside one of the translates of this parallelogram, and the quotientκ is one of its vertices. The remainder is the square distance fromα to this vertex, but the maximum possible distance in our algorithm is only${\displaystyle \frac{\sqrt{3}}{2}|\beta |}$, so. (The size ofρ could be slightly decreased by takingκ to be the closest corner.)

Ifx andy are Eisenstein integers, we say thatx dividesy if there is some Eisenstein integerz such thaty =zx. A non-unit Eisenstein integerx is said to be an Eisenstein prime if its only non-unit divisors are of the formux, whereu is any of the six units. They are the corresponding concept to theGaussian primes in the Gaussian integers.

There are two types of Eisenstein prime.

• an ordinaryprime number (orrational prime) which is congruent to2 mod 3 is also an Eisenstein prime.
• 3 and each rational prime congruent to1 mod 3 are equal to the normx² −xy +y² of an Eisenstein integerx +ωy. Thus, such a prime may be factored as(x +ωy)(x +ω²y), and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.

In the second type, factors of3,${\displaystyle 1-\omega }$ and${\displaystyle 1-{\omega }^2}$ areassociates:${\displaystyle 1-\omega =(-\omega )(1-{\omega }^2)}$, so it is regarded as a special type in some books.^[2][3]

The first few Eisenstein primes of the form3n − 1 are:

2,5,11,17,23,29,41,47,53,59,71,83,89,101, ... (sequenceA003627 in theOEIS).

Natural primes that are congruent to0 or1 modulo3 arenot Eisenstein primes:^[4] they admit nontrivial factorizations inZ[ω]. For example:

3 = −(1 + 2ω)²

7 = (3 +ω)(2 −ω).

In general, if a natural primep is1 modulo3 and can therefore be written asp =a² −ab +b², then it factorizes overZ[ω] as

p = (a +bω)((a −b) −bω).

Some non-real Eisenstein primes are

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

Up to conjugacy and unit multiples, the primes listed above, together with2 and5, are all the Eisenstein primes ofabsolute value not exceeding7.

As of October 2023^[update], the largest known real Eisenstein prime is thetenth-largest known prime10223 × 2^31172165 + 1, discovered by Péter Szabolcs andPrimeGrid.^[5] With one exception,^{[clarification needed]} all larger known primes areMersenne primes, discovered byGIMPS. Real Eisenstein primes are congruent to2 mod 3, and all Mersenne primes greater than3 are congruent to1 mod 3; thus no Mersenne prime is an Eisenstein prime.

The sum of the reciprocals of all Eisenstein integers excluding0 raised to the fourth power is0:^[6]$${\displaystyle \sum _{z\in \mathbf{E}\setminus \{0\}}\frac{1}{z^4}=G_4\left(e^{\frac{2\pi i}{3}}\right)=0}$$so${\displaystyle e^{2\pi i/3}}$ is a root ofj-invariant.In general${\displaystyle G_k\left(e^{\frac{2\pi i}{3}}\right)=0}$ if and only if${\displaystyle k\not\equiv 0(\mathrm{mod}6)}$.^[7]

The sum of the reciprocals of all Eisenstein integers excluding0 raised to the sixth power can be expressed in terms of thegamma function:$${\displaystyle \sum _{z\in \mathbf{E}\setminus \{0\}}\frac{1}{z^6}=G_6\left(e^{\frac{2\pi i}{3}}\right)=\frac{\mathrm{\Gamma }(1/3{)}^{18}}{8960{\pi }^6}}$$whereE are the Eisenstein integers andG₆ is theEisenstein series of weight 6.^[8]

Thequotient of the complex planeC by thelattice containing all Eisenstein integers is acomplex torus of real dimension 2. This is one of two tori with maximalsymmetry among all such complex tori.^[9] 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 ofGaussian 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
• Equianharmonic

• Eisenstein Integer--from MathWorld

• Algebraic numbers
• Quadratic irrational numbers
• Cyclotomic fields
• Lattice points
• Systolic geometry
• Integers

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