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.