Strong Goldbach Conjecture

The strong (binary) Goldbach conjecture asserts that every even integer $N\ge4$ is a sum of two primes. It originates in the 1742 correspondence between Goldbach and Euler [GoldbachEuler1742]; the letter used the then-common convention counting $1$ as a prime, and the statement above is its standard modern equivalent.

Two landmark partial results fall short in different ways. Chen proved that every sufficiently large even integer is the sum of a prime and a number with at most two prime factors [Chen1973Goldbach], one sieve-theoretic step from the conjecture. Helfgott completed the proof of the ternary (weak) Goldbach conjecture, that every odd integer greater than $5$ is a sum of three primes [Helfgott2015Ternary]. Neither route produces two genuine primes for every even $N$: the parity barrier in sieve theory obstructs upgrading Chen's almost-prime to a prime, and the ternary result concerns three summands.

The conjecture therefore remains open; a resolution must handle every even integer, not merely all sufficiently large ones, with both summands prime.