For any pair of integers m and n  , define K_m,n:={a²+mab+nb²|a,b∈Z}$K_{m,n}:=\{a^2+mab+n{b}^2|a,b\in \mathrm{Z}\}$.

11d9dcbad4adff69f2a5e382a7aa6" title="Click to view the MathML source">K_m,n$K_{m,n}$ is a semi-group with usual product of integers, for any pair of integers m and n.

A prime number p   can be expressed as p=a²±ab−b²$p=a^2\pm ab-b^2$ with integers a and b, if and only if, p is congruent to 0, 1 and −1 modulo 5.

A prime number p   can be expressed as p=a²±ab+b²$p=a^2\pm ab+b^2$ with integers a and b, if and only if, p is congruent to 0 and 1 modulo 3.