Determining the Solvability and Unsolvability of Quadratic Diophantine Equations Using Quadratic Residues and Continued Fractions

This paper determines the solvability and unsolvability of quadratic Diophantine equations using quadratic residues and continued fractions. Quadratic residues provide a framework for analyzing the conditions under which these equations have integer solutions, utilizing tools like the quadratic reciprocity law and the Legendre symbol. Continued fractions, particularly the periodic expansion of √D, provide a systematic approach to solving quadratic Diophantine equations like Pell’s equation x2-Dy2=N, where D is a positive, non-square integer and N is a non-zero integer. By combining these methods, we identify the solvability conditions and find integer solutions for quadratic Diophantine equations. The study also incorporates numerical approaches, supported by theorems, and demonstrates the results through illustrative examples.