Eisenstein integer

In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), sometimes called Eulerian integers (after Leonhard Euler), are special kinds of complex numbers. They look like this: z = a + bω, where a and b are regular whole numbers, and ω is a special number equal to (−1 + i√3)/2. This number ω is one of the cube roots of unity. This means that when you multiply it by itself three times, you get 1.

Eisenstein integers arrange themselves in a beautiful triangular lattice pattern when drawn on the complex plane. This is different from another group of numbers called Gaussian integers, which form a square lattice in the same plane. Both groups are made up of an endless number of points. This means there are countlessly many Eisenstein integers.

These numbers are important in many areas of math, especially in number theory and algebra. They help mathematicians study properties of numbers and solve equations in ways that aren’t possible with regular whole numbers alone. The pattern they form also appears in crystals and other natural structures. This shows how math connects to the world around us.

Properties

The Eisenstein integers are special numbers in mathematics. You can add, subtract, and multiply them like regular whole numbers. Each Eisenstein integer uses two whole numbers, a and b, and a special number called ω.

These numbers show interesting patterns when drawn on a graph. They make a triangle-shaped grid. Regular whole numbers make a square grid. This grid helps mathematicians study number theory and other parts of math.

Euclidean domain

The Eisenstein integers are a special kind of number system called a Euclidean domain. This means we can divide any two of these numbers and get a remainder that is smaller than the number we divided by.

Eisenstein primes are special Eisenstein integers that cannot be broken down into smaller pieces, except by multiplying by certain very simple numbers called units. Some regular whole numbers also act like Eisenstein primes. For example, numbers like 2, 5, 11, and others that leave a remainder of 2 when divided by 3 are Eisenstein primes. Other numbers, like 3 and some that leave a remainder of 1 when divided by 3, can be broken into smaller Eisenstein integers.

Eisenstein series

The Eisenstein series is a special type of math sum. An interesting fact is that if we add up the reciprocals of all Eisenstein integers (except zero) raised to the fourth power, the total equals zero.

We can also find the sum for the sixth power. This sum uses a function called the gamma function. The result is a special number, about 5.86303.

Quotient of C by the Eisenstein integers

The quotient of the complex plane C by the lattice of Eisenstein integers forms a complex torus, a special kind of shape. This torus has a high level of symmetry. You can imagine it by connecting the opposite edges of a regular hexagon.

Another torus with the same high symmetry is formed by the Gaussian integers. You can picture this by connecting opposite sides of a square.