In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known

The first few Eisenstein primes of the form are:

: 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... .

Natural primes that are congruent to or modulo are not Eisenstein primes: they admit nontrivial factorizations in . For example:

:

: .

In general, if a natural prime is modulo and can therefore be written as , then it factorizes over as

: .

Some non-real Eisenstein primes are

: , , , , , , .

Up to conjugacy and unit multiples, the primes listed above, together with and , are all the Eisenstein primes of absolute value not exceeding .

, the largest known real Eisenstein prime is the 12th-largest known prime , discovered by Péter Szabolcs and PrimeGrid.

Eisenstein series

The sum of the reciprocals of all Eisenstein integers excluding raised to the fourth power is :

<math display="block">\sum_{z\in\mathbf{E}\setminus\{0\\frac{1}{z^4}=G_4\left(e^{\frac{2\pi i}{3\right)=0</math>

so <math>e^{2\pi i/3}</math> is a root of j-invariant.

In general <math>G_k\left(e^{\frac{2\pi i}{3\right)=0</math> if and only if <math>k\not\equiv 0 \pmod 6</math>.

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

<math display="block">\sum_{z\in\mathbf{E}\setminus\{0\\frac{1}{z^6}=G_6\left(e^{\frac{2\pi i}{3\right)=\frac{\Gamma (1/3)^{18{8960\pi^6} \approx 5.86303</math>

where are the Eisenstein integers and is the Eisenstein series of weight 6.

Quotient of by the Eisenstein integers

The quotient of the complex plane by the lattice containing all Eisenstein integers is a complex torus of real dimension&nbsp;. This is one of two tori with maximal symmetry among all such complex tori. This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon.

thumb|center|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 .

See also

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

Notes

  • Eisenstein Integer--from MathWorld