Exploring the Infinitude of Primes and Related Conjectures

This paper presents several classical and modern proofs demonstrating the infinitude of prime numbers. The discussion includes Euclid’s original argument as well as alternative approaches including analytic, topological, and combinatorial proofs. In addition, this paper discusses two open problems in number theory: the infinitude of Mersenne primes and the Twin Prime Conjecture. The aim is to provide an overview of both established results and ongoing challenges in the study of prime numbers.