Browse › Number theory › 013 Ostmann’s inverse Goldbach conjecture
The additive indecomposability of the primes
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as $A+B$, where A and B are sets of nonnegative integers with at least two elements each.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Paolo Leonetti, On the inverse Goldbach problem (arXiv:2610.15758, 11 Oct 2026). Proves upper and lower counting-function bounds for a hypothetical decomposition of the primes and a lower bound for ternary sumsets; its introduction cites this paper and says the claim "has not yet been verified by the math community."
- Christian Elsholtz and Julia Stadlmann, The large sieve inequality and additive decompositions of sums of squares (arXiv:2610.20554, 14 Oct 2026). Proves a large-sieve estimate for sets that miss residue classes modulo prime squares, applies it to sums of two squares, and notes that OpenAI "claimed a full resolution of Ostmann's problem"; the authors state that their work predates the claim.
Formalization
- OpenAI math release, Ostmann’s inverse Goldbach conjecture: Lean scope note (lean/docs/013.md). Reports formalization of the full infinite-symmetric-difference statement for two nontrivial subsets of the nonnegative integers and of the two-infinite-summands case, with comparator files
OstmannComplete.leanandOstmannPrimes.lean. - google-deepmind/formal-conjectures, FormalConjectures/ErdosProblems/431.lean (formal-conjectures). States both the infinite-summands case and the full Ostmann statement, marked "research solved" and citing this paper; the theorem bodies are
sorryand link to the Lean proofs in the openai/math repository. - mo271, feat: mark problems solved in the OpenAI math release and link formal proofs (formal-conjectures PR #6898, merged 11 Oct 2026). Marks Erdős 431 as solved and adds a proof link for the Ostmann variant; the review discussion does not address this problem. [family-level]
- Shinsuke-Hamaji, ostmann-proof-lean4 (GitHub repository, 2026). A third-party repository that describes itself as a Lean 4 verification of Ostmann's conjecture that is "overriding OpenAI's frozen Result 013"; its relation to the release's Lean code is not documented.
Related prior literature mentioned by others
- Terence Tao and Tamar Ziegler, Infinite partial sumsets in the primes (arXiv:2301.10303, 2023). Cited on erdosproblems.com/431 as an unconditional construction of infinite sequences $A$ and $B$ with $a_i+b_j$ prime for $i<j$; this is a partial-sumset counterpart to the full-sumset statement proved here.
- Andrew Granville, A Note on Sums of Primes (Canad. Math. Bull. 33(4), 1990). Cited on erdosproblems.com/431 as a conditional result under the prime $k$-tuples conjecture: an infinite sequence in which the average of any two terms is prime.
- Ben J. Green and Adam J. Harper, Inverse questions for the large sieve (arXiv:1311.6176, 2013-11-24). The 数学ビジュアライズ exposition discusses its conditional inverse-sieve route to Ostmann’s conjecture as background to the release paper’s claimed approach without a general quadratic classification.
Possibly relevant literature (found by the agent, not checked)
- Imre Z. Ruzsa, Additive decomposition of signed primes (arXiv:2204.14013, 2022). Under the prime-tuple hypothesis, shows that the set of signed primes is a sumset; this is relevant to the paper's restriction to positive primes and summands in $\mathbb N_0$ (Theorem 1.1).
Digests and expositions
- 数学ビジュアライズ, OpenAIのAIが証明した「逆ゴールドバッハ予想」：素数は2つの集合の和に分けられない (数学ビジュアライズ, 2026-10-12). This Japanese exposition presents sumset examples, residue restrictions, the finite-summand reduction, and an outline of the manuscript’s proof following its introduction.
- Gradient Damage, Ostmann’s Inverse Goldbach Conjecture: The Primes Aren’t A + B (Gradient Damage, 2026-10-13). This short animation illustrates the claimed result through finite computations for two-element summands and explicitly distinguishes the illustration from a proof.
News and blogs
- Thomas Bloom, Erdős problems in the OpenAI proofs (erdosproblems.com blog, 8 Oct 2026). Has a section headed "Primes are not a sumset" that summarizes the claimed result as Problem 431, notes that the proof has been formalised, and says the PDF is 80 pages. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 013
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:the-additive-indecomposability-of-the-primes-September-24-2026,
author = {{OpenAI}},
title = {{The additive indecomposability of the primes}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf}{OAI:the-additive-indecomposability-of-the-primes-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/013.md. The Lean statements have not been checked by people against the paper's claims.
Family 013: Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as $A+B$ with $A,B\subseteq\mathbb Z_{\ge0}$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.
This is the only paper in this family.