The conjecture states that every even integer 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 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.