Goldbach Conjecture

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

Marginal Conjecture
Every integer greater than $5$ can be written as the sum of three primes.

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

Comment
This problem is the second part of no. 8 in the Hilbert 23.

Also see

 * Goldbach's Weak 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).

The two conjectures have been shown to be equivalent.