EYLER VA FERMA TEOREMASI

Аннотация

This article presents a comprehensive exploration of two key theorems in number theory: Fermat’s Little Theorem and Euler’s Theorem. Both theorems are essential tools in modular arithmetic and play a significant role in simplifying complex operations involving prime numbers and integer relations. The article begins by explaining the core concepts, mathematical formulations, and proofs of each theorem. Their connections, differences, and generalizations are then discussed in detail. Furthermore, the article examines practical applications of these theorems through illustrative problem-solving examples involving modular equations and arithmetic operations. Special attention is given to their role in the field of cryptography, particularly within the RSA encryption algorithm, which forms the https://scientific-jl.com 3 8 61-son_3-to’plam_May-2025 ISSN:3030- 3 6 1 3 T A D Q I Q O T L A R jahon ilmiy – metodik jurnali backbone of modern digital security. The paper highlights how these theoretical concepts extend far beyond pure mathematics and are actively used in real-world applications such as secure communication and data protection. The article serves as a valuable resource for students, educators, and researchers aiming to deepen their understanding of number theory and its practical utility. Kalit so‘zlar: Ferma kichik teoremasi, Eller teoremasi, sonlar nazariyasi, modulli arifmetika, tub son, kriptografiya, RSA algoritmi, matematik isbot, modulli tenglama, raqamli xavfsizlik Keywords: Fermat's Little Theorem, Euler’s Theorem, number theory, modular arithmetic, prime number, cryptography, RSA algorithm, mathematical proof, modular equation, digital security