The Goldbach Conjecture

Overview & Synopsis

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In order to prove this statement, we begin by introducing a kind of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift out from $[1, x]$ all those elements that are congruents to $0$ modulo $p$, or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. So, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. Using this tool, we prove that there exists an integer $K_\alpha > 5$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$. This result implies that every even integer $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes. Furthermore we provide the upper estimation $K_\alpha < 89$.