What Are Prime Numbers

Prime numbers are positive integers who have exactly two distinct divisors- 1 and itself. Every positive integer can be uniquely factorised as a product of powers of distinct prime numbers. Euler proved that there were infinitely many prime numbers. 2 is the only even prime number. 5 is the only prime number which ends in 5.

There is no explicit formula for the sequence of prime numbers. However, there exists a prime number theorem which states that for prime counting function p(x), the asymptotic behaviour of p(x) closely resembles x/log(x). The Riemann hypothesis conjects that the zeros of the Riemann zeta function line on the line Real(s)=1/2 where Real(s) denotes the real part of s. If this hypothesis is proven, then for logarithmic integral function li(x)- the integral of 1/ln(x)- the prime counting function p(x) and li(x) are closely related by the zeros of the Zeta function. I.e. magnitude of error between primes and their expected position via the li(x) formula is controlled by the real part of the zeros of the Zeta function.

I myself have attempted to look into this problem, looking from a Further Complex Methods and Integrable systems point of view. My progress is very much dependent on finding the best Lie symmetry for an equation (equivalent of finding a needle in a haystack!).