The distribution of prime numbers has been the object of intense study by many modern mathematicians. Gauss and Legendre conjectured the prime number theorem which states that the number of primes less than a positive number $$x$$ is asymptotic to $$x/\log x$$ as $$x$$ approaches infinity. This conjecture was later proved by Hadamard and Poisson. Their proof and many other proofs lead to what is known as Analytic Number theory. In this chapter we demonstrate elementary theorems on primes and prove elementary properties and results that will lead to the proof of the prime number theorem.