EUCLID’S THEOREM AND ITS APPLICATIONS IN NUMBER THEORY AND MODERN COMPUTATION

Abstract

Classical number theory contains several foundational principles that continue to influence modern mathematics and computer science. Among these principles, the theorem associated with the ancient Greek mathematician Euclid provides a systematic method for analyzing divisibility and determining the greatest common divisor of integers. The theorem forms the theoretical basis of the Euclidean algorithm, one of the oldest and most efficient algorithms in mathematics. Over time, the ideas derived from Euclid’s work have become essential in fields such as algebra, computational mathematics, and modern cryptography. The present study examines Euclid’s theorem through a formal theorem–proof framework, demonstrating its logical structure and its applications in number theory, including Euclid’s lemma and the proof of the infinitude of prime numbers. The discussion also highlights the relevance of these classical ideas in contemporary computational systems.