# Goldbach Conjecture

**Type:** Conjecture  
**Domain:** Number Theory  
**Posed:** 1742, Christian Goldbach (letter to Euler)  
**Status:** OPEN (strong/binary form); weak/ternary form PROVEN (Helfgott, 2013)  
**Codex URL:** /mathematics/number-theory/goldbach-conjecture/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Every even integer >2 is the sum of two primes. Verified to 4×10¹⁸, still unproven.

## Strong vs weak Goldbach
Strong (binary): every even n>2 = sum of 2 primes — OPEN.
Weak (ternary): every odd n>5 = sum of 3 primes — PROVEN (Helfgott 2013, building on Vinogradov 1937).

## Chen's theorem (1973)
Every sufficiently large even number = prime + semiprime (product of ≤2 primes) — falls short of full conjecture.

## Why it's hard
Primes are multiplicatively defined; Goldbach is an additive question — a recurring source of difficulty in analytic number theory.

## Hardy-Littlewood circle method
Key technique (1920s) underlying Helfgott's proof and partial progress toward the strong conjecture.

## Sources
### Tier 1
- Helfgott, H.A. (2013). arXiv:1312.7748.
- Chen, J.R. (1973). Scientia Sinica.
### Tier 2
- Oliveira e Silva, T. et al. (2014). Mathematics of Computation.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*