Let k ≥ 1 be an integer and let P = k + 2, Q = k and D = k 2 + 4. In this paper, we derived some algebraic properties of quadratic ideals Iγ and indefinite quadratic forms Fγ for quadratic irrationals γ, and then we determine the set of all integer solutions of the Diophantine equation F ±k γ (x, y) = ±Q.

In this work, we deduce some algebraic relations on base points [Formula: see text] of positive definite forms [Formula: see text] and [Formula: see text]-basis of [Formula: see text].