Goldbach Conjecture

Conjecture
Every even integer greater than $2$ is the sum of two primes.

Progress
It has been checked by computer for numbers up to at least $10^{18}$.

Partial results are given by Chen's Theorem and Vinogradov's Theorem.

Also see

 * Goldbach's Weak Conjecture
 * Goldbach Conjecture implies Goldbach's Marginal Conjecture

It was he who made what is now known as Goldbach's Marginal Conjecture in 1742 in a letter to. then proposed this stronger conjecture.