The conjecture states that every even integer $n>2$ is expressible as the sum of two primes.

In 1966 Chen proved that every sufficiently large even number can be expressed as the sum of a prime and a number with at most two prime divisors.

Vinogradov proved that every sufficiently large odd number is a sum of three primes. In 1997 it was shown by J.-M. Deshouillers, G. Effinger, H. Te Riele, and D. Zinoviev that, assuming a generalized Riemann hypothesis, every odd number $n>5$ can be represented as sum of three primes.

The conjecture was first proposed in a 1742 letter from Christian Goldbach to Euler and still remains unproved.