Goldbach's weak conjecture

The problem asserts that every odd integer above $5$ is a sum of three primes, with the stronger odd-prime version for integers above $7$. Harald Helfgott announced the complete proof in $2013$; later sources distinguish this settled ternary statement from the still-open strong Goldbach conjecture.

Known results

• Vinogradov proved the assertion for sufficiently large odd integers; later estimates gave a threshold near $10^{1346}$.
• Yannick Saouter checked the conjecture computationally through $10^{20}$.
• Terence Tao proved that every odd integer is a sum of at most five primes, in $2012$.
• Helfgott reduced the analytic threshold to roughly $10^{27}$-$10^{30}$, with David Platt verifying the finite remainder.

2013-2015 proof and subsequent confirmation

Helfgott’s proof combined explicit circle-method estimates with computation and established the full range. It was accepted for the Annals of Mathematics Studies book series in $2015$; later papers state the theorem as proved, and a $2024$ paper develops a restricted-prime extension rather than challenging the original result.

Current status (as of August 2026): The weak Goldbach conjecture, including the odd-prime formulation for odd integers above $7$, is resolved; the strong Goldbach conjecture remains open.