# Weak Goldbach conjecture

- kind: problem
- status: solved (evidence tier V3)
- year: 1742
- authors: from the Goldbach–Euler correspondence

**Statement.** Every odd integer greater than 5 is the sum of three primes.

Vinogradov proved in 1937 that it holds for all sufficiently large odd numbers, but the threshold was astronomically large. Helfgott sharpened the circle method until the threshold fell to a range that a computer had already checked. It shows a common endgame: an asymptotic proof, then decades of bookkeeping to make its constants small.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1742, from the Goldbach–Euler correspondence. Closed 2013, Harald Helfgott.

## Sources

- [Wikipedia: Goldbach's weak conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture)

[Number theory](/wiki/Space:number-theory.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- used by [goldbach-conjecture](/wiki/Problem:goldbach-conjecture.md) — Progress here drew on Weak Goldbach conjecture, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.
- related to [riemann-hypothesis](/wiki/Problem:riemann-hypothesis.md) — The original curation links this problem to Weak Goldbach conjecture; the argument for the link has not been written yet.

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Revision 7, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:weak-goldbach?action=history. Edit: POST /api/wiki/page/Problem:weak-goldbach {source, summary, base_rev: 7}.