Abstract
Given a binary quadratic polynomial \(f(x_1,x_2)=\alpha x_1^2+\beta x_1x_2+\gamma x_2^2\in \mathbb {Z}[x_1,x_2]\), for every \(c\in \mathbb Z\) and \(n\ge 2\), we study the number of solutions \(\mathrm {N}_J(f;c,n)\) of the congruence equation \(f(x_1,x_2)\equiv c\bmod {n}\) in \((\mathbb {Z}/n\mathbb {Z})^2\) such that \(x_i\in (\mathbb {Z}/n\mathbb {Z})^\times \) for \(i\in J\subseteq \{1,2\}\).
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Acknowledgements
Research is partially supported by National Natural Science Foundation of China (Grant No. 11571328).
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Liu, Y., Ouyang, Y. On Binary Quadratic Forms Modulo n. Commun. Math. Stat. 7, 61–67 (2019). https://doi.org/10.1007/s40304-018-0141-1
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DOI: https://doi.org/10.1007/s40304-018-0141-1