Quadratic diophantine equations and orders in quaternion algebras

Abstract

We first reformulate the main theorem on quadratic Diophantine equations given in our recent book in terms of the even Clifford group. In the quaternary case the group is the multiplicative group of a quaternion algebra, and the reformulated theorem connects a class of primitive solutions of a quadratic equation with a class of an order in the algebra. In the indefinite case each class is essentially a class of binary hermitian forms. In the definite case the result is an analogue of the result of Gauss on the sums of three squares.