Closing the gap: the ternary Goldbach conjecture, unconditionally (Helfgott, 2013)

Frequently asked questions

Does this proof say anything about the still-open binary Goldbach conjecture?

No. Matching a sum to exactly two primes instead of three removes an entire degree of freedom from the circle-method computation, requiring minor-arc control far beyond what current methods provide. The ternary case is not a stepping stone that mechanically extends to the binary case; the two remain genuinely different problems.

Source

Harald A. Helfgott (with numerical verification joint with David J. Platt), 2013

The ternary Goldbach conjecture is true

arXiv:1312.7748v1

Preprint

Condensed summary

Step 5 of 5: Closing the loop: numerical verification meets the analytic threshold

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In plain words

The analytic argument of Steps 2–4 settles every odd $n\ge 10^{27}$ — still far too large to check number by number, but now within reach of a different kind of computation: verifying that every even number up to some bound is a sum of two primes (the classical Goldbach conjecture, checked far beyond what is needed here) is enough, because $n=2+({\rm even})$ or a short additional argument reduces small odd cases to it. Helfgott and David Platt carried out a rigorous, computer-assisted verification reaching $8.875\times10^{30}$ — comfortably past the $10^{27}$ threshold — so the two ranges overlap with room to spare.

10^{27} \le n \le 8.875\times 10^{30} : \text{ verified by computer (Helfgott–Platt, 2013)} \quad \Longrightarrow \quad \forall\, n>5 \text{ odd}: n=p_1+p_2+p_3

Detailed analysis

Concretely, $8{,}875{,}694{,}145{,}621{,}773{,}516{,}800{,}000{,}000{,}000$ is the exact numerical bound Helfgott and Platt (2013) reached, comfortably exceeding $10^{27}$ . Since every odd $n\ge10^{27}$ is covered analytically (Steps 2–4) and every odd $5<n\le 8.875\times10^{30}$ is covered computationally, and $8.875\times10^{30}>10^{27}$ , the two ranges overlap and jointly cover every odd $n>5$ with no gap — establishing the ternary Goldbach conjecture as an unconditional theorem. The full argument, spanning several long papers (2012–2013) and requiring an unusually extended, multi-round refereeing process, was later consolidated into a research monograph in the Annals of Mathematics Studies series.