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ADVANCES IN DIOPHANTINE EQUATIONS: EXPLORING RATIONAL POINTS ON ELLIPTIC CURVES

ABSTRACT

Number theory, a branch of mathematics with deep historical roots, continues to be a fertile ground for ground breaking discoveries. This paper contributes to the ongoing exploration of Diophantine equations, focusing on the study of rational points on elliptic curves, a fundamental area of interest in number theory. Our research begins by providing an accessible introduction to Diophantine equations, outlining the importance of finding rational solutions to these equations, and their relevance in various mathematical disciplines, including cryptography and algebraic geometry. We delve into the background of elliptic curves, emphasizing their role as essential objects of study in modern number theory. In the first section of our paper, we introduce recent advances in the theory of elliptic curves. We present a comprehensive overview of the Mordell-Weil Theorem and its implications for the structure of rational points on elliptic curves. We also discuss the significance of the Birch and Swinnerton-Dyer Conjecture in the context of rank computations. The second section focuses on specific techniques and algorithms for finding rational points on elliptic curves. We explore various computational methods, including the use of 2-descent, Selmer groups, and descent via isogeny, highlighting their applicability and limitations in practice. In the third section, we present original research findings, where we investigate rational points on a selected set of elliptic curves over different number fields. Our study involves both theoretical and computational approaches, providing insights into the distribution and behavior of rational points. We also discuss applications of our results in cryptography, particularly in the design of secure elliptic curve-based cryptographic schemes. The final section of our paper outlines open questions and directions for future research in this dynamic field of number theory. We encourage further exploration of the interaction between elliptic curves, modular forms, and L-functions, as well as the development of improved algorithms for finding rational points. In conclusion, this research paper contributes to the ongoing advancement of number theory, specifically in the realm of Diophantine equations and rational points on elliptic curves. We hope that our work serves as a valuable resource for mathematicians, researchers, and students interested in this captivating area of mathematical inquiry.

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