The problem
Every odd integer greater than 5 can be written as the sum of three primes (repetitions allowed).
The Weak Goldbach Conjecture (ternary)
historicReferences
History & significance
Implicit in the 1742 Goldbach–Euler correspondence. Vinogradov (1937) removed reliance on GRH for all sufficiently large odds, but "sufficiently large" meant astronomically beyond reach; explicit bounds descended from e^{e^{\(e^{41}\)}} toward computability over decades (Liu–Wang reached \(e^{3100}\)).
Connected problems
The resolution (human proof)
Proved completely by Harald Helfgott, 2013: sharpened circle-method estimates handle all n ≥ 10²⁷, with exhaustive computation covering the remainder below — the two halves finally meeting. A reminder that even "weak" forms of classical conjectures can take centuries; its big sibling, the even/binary Goldbach conjecture, remains on our unsolved shelf.