@article{Zhang2026, 
author = {Jiafan Zhang and Ran Xiong},
title = {Explicit formulae for inhomogeneous binary quadratic congruences modulo prime powers},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {10},
pages = {7046-7070},
keywords = {binary quadratic congruences, exponential sums, quadratic Gauss sums, prime powers, Hensel lifting},
url = {https://www.sciopen.com/article/10.3934/era.2026305},
doi = {10.3934/era.2026305},
abstract = {In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence    a      x    2    +  b  x  y  +  c      y    2    +  d  x  +  e  y  ≡  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              (                        Z                q            )        2   and the number of solutions with both variables restricted to units over              (                        Z                q        ∗            )        2  . This paper simultaneously handles a mixed quadratic term and two separate linear terms, extending several previously studied diagonal, homogeneous, and symmetric special cases.}
}