The problem
Every odd integer \(n > 5\) is the sum of three odd primes (repetitions allowed), e.g. \(7 = 2+2+3\). Vinogradov (1937) proved it for all sufficiently large odds; Helfgott (2013) closed the remaining range, completing the proof.
The Weak Goldbach Conjecture (ternary)
historicHistory & 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
- special case of Goldbach's Conjecture
More in number theory
- same area Dickson's Conjecture
- same area Schinzel's Hypothesis H
- same area The Existence of Infinitely Many Fermat Primes
References
- primaryRead the primary source →
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.
Verified by curator — see the API for full claim provenance.