Diophantine equations constitute a fundamental area of number theory characterized by the requirement that solutions must be integers. These equations have played a central role in mathematical research from classical antiquity to modern computational science. This paper presents a systematic study of the mathematical foundations and computational techniques associated with Diophantine equations. Core theoretical concepts—including divisibility theory, greatest common divisors, modular arithmetic, and prime factorization—are examined to establish existence and structure conditions for integer solutions. Classical solution methods such as the Euclidean and Extended Euclidean algorithms are discussed alongside contemporary computational approaches that utilize symbolic computation and algorithmic search strategies. The study further explores diverse applications of Diophantine equations in cryptography, coding theory, optimization, computer science, and geometric modeling. By integrating theoretical analysis with modern computational tools, this work highlights the continued relevance of Diophantine methods in both pure mathematics and applied technological domains. The findings emphasize the importance of efficient algorithm design and suggest future research directions in higher-degree equations, artificial intelligence–assisted number theory, and large-scale computational analysis.
Yadav et al. (Thu,) studied this question.