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
 * Equivalence of Goldbach Conjecture and Goldbach's Marginal Conjecture

He first made this conjecture in 1742 in a letter to Euler. The marginal conjecture was written in the margin of this letter (hence its name).