Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown (by computerised testing) to hold for all integers less than 4 × 10¹⁸, but remains unproven for all possible numbers.

Further reading courtesy Mathematics and Statistics dept. University Tennessee at Martin, US

Goldbach's original letter to Euler in 1742.

Note that Goldbach's weak conjecture (a.k.a Goldbach's odd conjecture)

Every odd number greater than 7 can be expressed as the sum of three odd primes

is generally accepted to have been finally proved in 2013, by Harald Helfgott