In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence $ ax^2+bxy+cy^2+dx+ey\equiv n\pmod{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^\ast\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.
Citation: Jiafan Zhang, Ran Xiong. Explicit formulae for inhomogeneous binary quadratic congruences modulo prime powers[J]. Electronic Research Archive, 2026, 34(10): 7046-7070. doi: 10.3934/era.2026305
In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence $ ax^2+bxy+cy^2+dx+ey\equiv n\pmod{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^\ast\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.
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Jiafan Zhang, Ran Xiong. Explicit formulae for inhomogeneous binary quadratic congruences modulo prime powers[J]. Electronic Research Archive, 2026, 34(10): 7046-7070. doi: 10.3934/era.2026305
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