RIEMANN AND DIOPHANTINE’S CONTRIBUTION IN THE FIELD OF NUMBER THEORY

Abstract

There have been several fascinating applications of Number Theory in key cryptography. Key cryptography enables many technologies we take for granted, such as the ability to make secure online transactions. The purpose of this survey paper is to highlight certain important such applications. Prime numbers constitute an interesting and challenging area of research in number theory. Diophantine equations form the central part of number theory. An equation requiring integral solutions is called a Diophantine equation. In the first part of this paper, some major contribution in number theory using prime number theorem is discussed and some of the problems which still remains unsolved are covered. In the second part some of the theorems and functions are also discussed such as Diophantine Equation, Goldbach conjecture, Fermats Theorem, Riemann zeta function and his hypothesis that still remain unproved to this day . The Chinese hypothesis is a special case of Fermat’s little theorem. As proved later in the west, the Chinese hypothesis is only half right . From the data of this study we conclude that number theory is used in computer network and applications in cryptography. We came to know about the purpose of Diophantine equation , Square-free natural number, Zeta function, Fermat’s theorem and Chinese hypothesis which is a special case for Fermat’s theorem.