Geometric approach to the Diophantine equation x^2 + x y - y^2 = m

We investigate the Diophantine equation x^2 + x y - y^2 = m, for m ∈ Z given, from a geometric point of view. The hyperbola given by the equation carries a known group structure, which we interpret in four different ways, firstly with the familiar parallel line construction. It turns out that the group defined by this construction corresponds to the restriction of a group operation on R^2, which is induced by the number field Q( √ 5). In this way, the group operation can be described using lean formulae. We also find a parameterisation of the hyperbola that is compatible with the group operation. This result is analogous to the fact that the parameterisation of a cubic curve using the Weierstrass ℘-function is compatible with the group structure of the elliptic curve. A fourth interpretation of the group structure is based on a geometric observation about the orientated area of triangles with one vertex in the origin and two vertices on the hyperbola. Our parameterisation allows to define intervals whose images are regions F_m on the hyperbola such that any integer solution is uniquely given by applying a linear map to ( x, y ) ∈ F_m. We also expand the considerations for solvability of Cohn of binary quadratic forms obtained from quadratic number fields and give descriptions of the integer solutions to x^2 + x y - y^2 = m using the ideals in the number field Q( √ 5).