Definitive Proof of Goldbach's conjecture

Abstract

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture was first proposed by German mathematician Christian Goldbach in 1742 and, despite being obviously true, has remained unproven. In this paper, it is shown that the set of all even integers n that are not divisible by a prime number less than the square root of n has the relatively fewest number of prime pairs. An equation was derived that approximates the number of prime pairs for these values of n. It was then proven that this equation never goes to zero for any n, and as n increases, the number of prime pairs also increases, thus validating Goldbach's conjecture. Error analysis was performed to show that the difference between this approximation and the actual number of prime pairs is small enough so that for all n > 622, the number of prime pairs of n is greater than 1, thus proving Goldbach's conjecture.