Once Upon A Prime

A fundamental theorem in arithmetic states that every composite number is a product of primes ,and moreover, in a unique way. This book deals with important questions relating to primes: how to decide whether an arbitrary given natural number (>1) is prime or not? In chapter one I provide with some efficient simple tests for primality, although it is well known that deciding whether a given natural number is prime or composite, is not an easy task .The Riemann zeta function is important in the theory of primes In this book, I prove a hypothesis based on the Riemann zeta function known as the Riemann hypothesis. A proof of the theorem which states that there exists at least one prime between any natural number n (>1) and its' double 2n ,was supplied by the Indian genius Ramanujan. I give in chapter 3 a proof of a similar theorem which asserts that ,if n is a natural number (>1), then there is always a prime between the square on and the square of its' successor (n+1) .