An Eisenstein triple is a set of three positive integers $(a,,b,,c)$ that satisfy the Diophantine equation
$$ a^{2} - a b + b^{2} = c^{2}, $$
which is the norm equation for the ring of Eisenstein integers $\mathbb{Z}[\omega]$ where $\omega = e^{2\pi i/3}=(-1+\sqrt{-3})/2$. The equation can be interpreted as the condition that the Eisenstein integer $a + b\omega$ has Euclidean norm $c$. In this sense an Eisenstein triple plays a role analogous to that of a Pythagorean triple, which satisfies $a^{2}+b^{2}=c^{2}$ and corresponds to the norm in the Gaussian integers $\mathbb{Z}[i]$.
Parametrization
All primitive Eisenstein triples (those with $\gcd(a,b,c)=1$ and not a multiple of a smaller triple) can be generated by two coprime integers $m$ and $n$ of opposite parity, using the formulas
$$ \begin{aligned} a &= |m^{2} - n^{2}|,\ b &= 2mn - n^{2},\ c &= m^{2} - mn + n^{2}, \end{aligned} $$
or any permutation of $(a,b)$. Multiplying a primitive triple by a positive integer $k$ yields a non‑primitive (or “scaled”) Eisenstein triple $(ka,,kb,,kc)$.
Examples
- $(1,,1,,1)$ – the smallest Eisenstein triple.
- $(3,,5,,7)$ because $3^{2} - 3\cdot5 + 5^{2}=9 -15 +25 =19 = 7^{2}$.
- $(7,,13,,15)$ since $7^{2} - 7\cdot13 + 13^{2}=49 -91 +169 =127 = 15^{2}$.
Algebraic and Geometric Interpretation
In the complex plane, the points representing the Eisenstein integers form a triangular lattice. The condition $a^{2} - a b + b^{2}=c^{2}$ asserts that the distance from the origin to the lattice point $a + b\omega$ equals the integer $c$. Consequently, Eisenstein triples describe integer‑length vectors in this lattice.
Historical Context
The terminology derives from Ferdinand Eisenstein (1823–1852), a German mathematician noted for his work on algebraic number theory and the introduction of Eisenstein integers. The systematic study of triples satisfying the Eisenstein norm equation parallels the classical treatment of Pythagorean triples dating back to Euclid and Diophantus.
Applications
Eisenstein triples appear in problems concerning tilings of the plane by equilateral triangles, in the classification of integer solutions to certain quadratic forms, and in cryptographic constructions that exploit the arithmetic of $\mathbb{Z}[\omega]$.
Related Concepts
- Pythagorean triple – integer solutions of $a^{2}+b^{2}=c^{2}$.
- Gaussian integer – complex numbers of the form $x+yi$ with integer $x, y$.
- Eisenstein integer – complex numbers of the form $x + y\omega$ with integer $x, y$.
- Norm form – the quadratic form associated with the norm in a quadratic integer ring.
References
- N. J. Fine, Algebraic Number Theory, Springer, 1991 – discussion of norms in quadratic integer rings.
- H. S. M. Coxeter, Introduction to Geometry, Wiley, 1969 – geometric interpretation of Eisenstein lattices.
- R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 – includes a section on Eisenstein triples.