Elementary Number Theory: Foundations, Applications, And Recent Developments

Elementary number theory is a key branch of mathematics focuses on integer properties and relationships. This review comprehensively covers the key concepts, theorems, and applications. It examines integer properties like divisibility, primality, and congruences, and introduces the division and Euclidean algorithms as essential tools. The paper explores prime numbers, their infinitude, and the prime number theorem. The fundamental theorem of arithmetic, stating that every positive integer has a unique prime factorization, is discussed with its proof and significance. Diophantine equations, polynomial equations involving integers, are investigated with solutions methods. Applications in various fields are highlighted, including the RSA algorithm and DiffieHellman key exchange in cryptography, error-correcting codes like Hamming and Reed-Solomon in coding theory, and algorithm studies in computer science. This review is a valuable resource for students and researchers of elementary number theory and its modern implications