Investigating Integer Solutions to Quadratic Diophantine Equations Using Quadratic Residues

Abstract

In this paper, we study integer solutions to quadratic Diophantine equations of the form $x^{2} - Dy^{2} = N$, where $D$ is a composite number that is not a perfect square, such as $D = 14, 15, 18$, and $N$ is an odd integer. We discuss the quadratic residues and use some strategies to determine whether these equations are solvable or not. Additionally, we apply strategies, such as the Euclidean algorithm, B$\acute{e}$zout's identity, Thue's theorem, the Legendre symbol, the quadratic reciprocity law, and the Chinese remainder theorem to analyze the solvability and unsolvability of modified quadratic Diophantine equations. We identify research gaps in solving quadratic Diophantine equations, particularly when $D$ is a large composite number and $N$ is a large integer, and develop computational methods to enhance their solvability.