The Goldbach conjecture originates from a $1742$ letter from Goldbach to Euler, states that every even integer greater than $3$ is the sum of two primes. The conjecture has been shown to hold for all integers less than $4 × 10^{18}$, but remains unproven despite considerable effort.
Goldbach's weak conjecture is that every odd integer greater than $6$ is the sum of three primes.