Application of Special Number's in Number Theory

Akshit Chauhan, Manu Gupta

Abstract: This paper presents a focused study on two essential categories of numbers in number theory: Prime Numbers and Fermat Numbers. Prime numbers serve as the foundational elements of arithmetic and are widely used in cryptographic systems due to their unique factorization properties. Fermat numbers, defined as Fn = 2^(2^n) + 1, exhibit rare and intriguing mathematical characteristics, including coprimality and applications in polygon construction. While primes are central to both theory and modern encryption, Fermat numbers challenge computational limits in factorization and primality testing. This work explores their properties, theorems, applications, and unresolved questions in mathematical research.

Keywords: Prime Numbers, Fermat Numbers, Cryptography, Public-key Cryptography, RSA Algorithm, Modular Arithmetic, Primality Testing, Fermat?s Little Theorem, Wilson?s Theorem, Coprimality, Generalized Fermat Numbers

How to Cite?: Akshit Chauhan, Manu Gupta, "Application of Special Number's in Number Theory", Volume 14 Issue 7, July 2025, International Journal of Science and Research (IJSR), Pages: 1684-1687, https://www.ijsr.net/getabstract.php?paperid=MR25727204304, DOI: https://dx.doi.org/10.21275/MR25727204304