Zero-Free Primitive Square Representations by Some Ternary Quadratic Forms

Abstract

The number of distinct zero-free primitive square representations by the ternary quadratic formsis expressed in terms of the number of primitive square representations by this form taking into account zeros, permutations and sign changes, and the number of distinct primitive square representations by certain binary quadratic forms. Asymptotic expressions for the cumulative numbers are also derived and the obtained formulas are illustrated at two typical cases.