Explicit formulae for inhomogeneous binary quadratic congruences modulo prime powers

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Abstract

In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence ax² + bxy + cy² + dx + ey ≡ n (mod q), where q is a prime power. We treat the cases where q is an odd prime power and a power of 2 separately. By combining exponential sum methods, quadratic Gauss sums, and Hensel’s lemma, we obtain exact formulae for the number of unrestricted solutions over $\left(\mathbb{Z}_{q}\right)^{2}$ and the number of solutions with both variables restricted to units over $\left(\mathbb{Z}_{q}^{*}\right)^{2}$. This paper simultaneously handles a mixed quadratic term and two separate linear terms, extending several previously studied diagonal, homogeneous, and symmetric special cases.

Keywords

binary quadratic congruences / exponential sums / quadratic Gauss sums / prime powers / Hensel lifting

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