Similar to a Pythagorean triple, an Eisenstein triple (named after the German mathematician Gotthold Eisenstein) is a set of three positive integers that are the lengths of the sides of a triangle where one of the angles is 60° or 120°. The relationship of such triangles to the Eisenstein integers is analogous to the relationship of Pythagorean triples to the Gaussian integers.
Triangles with an angle of 60°
For a triangle with an angle of 60°, the Law of Cosines gives the relationship:
$$ c^2 = a^2 - ab + b^2 $$
where $c$ is the side opposite the 60° angle. When $a$, $b$, and $c$ are positive integers satisfying this equation with $a < c < b$, the triple $(a, b, c)$ is called an Eisenstein triple.
Examples of Eisenstein triples (60° case) include:
| Side $a$ | Side $b$ | Side $c$ |
|---|---|---|
| 3 | 8 | 7 |
| 5 | 8 | 7 |
| 5 | 21 | 19 |
| 7 | 15 | 13 |
| 7 | 40 | 37 |
| 8 | 15 | 13 |
| 9 | 24 | 21 |
Triangles with an angle of 120°
For a triangle with an angle of 120°, the Law of Cosines gives:
$$ c^2 = a^2 + ab + b^2 $$
Examples of such triples include:
| Side $a$ | Side $b$ | Side $c$ |
|---|---|---|
| 3 | 5 | 7 |
| 7 | 8 | 13 |
| 5 | 16 | 19 |
Relationship to Eisenstein integers
Eisenstein triples are closely connected to the ring of Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega = e^{2\pi i/3} = (-1 + i\sqrt{3})/2$ is a primitive cube root of unity. The norm of an Eisenstein integer $z = a + b\omega$ is given by:
$$ N(a + b\omega) = a^2 - ab + b^2 $$
Thus, if $(a, b, c)$ is an Eisenstein triple, then $N(a + b\omega) = c^2$, meaning the norm of the Eisenstein integer $a + b\omega$ is a perfect square.
Properties
- A primitive Eisenstein triple is one where $\gcd(a, b, c) = 1$.
- If $(a, b, c)$ is a primitive Eisenstein triple, then $(b - a, b, c)$ is also a primitive Eisenstein triple (called its conjugate).
- The hypotenuse $c$ in a primitive Eisenstein triple must have all prime factors congruent to 1 modulo 3.
- The number of primitive Eisenstein triples with a given hypotenuse $c$ (where all prime factors of $c$ are $\equiv 1 \pmod{3}$) is $2^k$, where $k$ is the number of distinct prime factors of $c$.
See also
- Loeschian number
- Integer triangles with a 60° angle
- Integer triangles with a 120° angle
- Eisenstein integer
