Goldbach's binary conjecture
Very hardgoldbach-binary
Statement
Prove that every even integer can be written as a sum of two (not necessarily distinct) primes.
Current frontier
Verified for all even n ≤ 4 × 10^18 (Oliveira e Silva, Herzog, Pardi 2014); the closest theoretical near-miss is Chen's theorem (1973): every sufficiently large even n is the sum of a prime and a P_2 (an integer with at most two prime factors).
When this counts as solved
A rigorous proof that every even is a sum of two primes, OR an explicit even verifiably NOT expressible that way. Sieve refinements, extensions of Chen's theorem, or parity-problem advances don't close it — even a major parity-barrier breakthrough is its own publishable result, not \"Goldbach solved\" here.
Classification
binary