FOR DIFFERENT PRIME NUMBERS D AND AN ODD INTEGER N IN THE QUADRATIC DIOPHANTINE EQUATION X² − DY² = N

In this paper, we study the integral solutions to the quadratic Diophantine equation of the form X² − DY² = N, where D is a prime number, especially D = 13, 17, 19, and N is an odd integer. We describe the concepts of quadratic residues and use algebraic methods to determine the solvability or unsolvability of quadratic Diophantine equations. Moreover, we derive significant results on the solvability or unsolvability of modified quadratic Diophantine equations with varying values of D and N, using various mathematical tools such as the Euclidean algorithm, Thue’s Theorem, and the Chinese Remainder Theorem. Our results enhance the understanding of the relationship between prime numbers, odd integers, and the structure of solutions to the quadratic Diophantine equation.