Result 013, Number theory

Ostmann’s inverse Goldbach conjecture

Proves that no finite modification of the primes can be written as A+BA+B with A,B⊆Z≥0A,B\subseteq\mathbb Z_{\ge0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.

Lean formalization Proof

The bigger picture

Why it matters

Can the primes be assembled by adding every element of one set to every element of another? The manuscript claims this is impossible for two nontrivial sets, revealing a limit on the primes' additive structure.

What changes?

The manuscript reports a proof of Ostmann's inverse Goldbach conjecture. Its claim concerns every set obtained from the primes by adding or removing finitely many numbers. None can equal a sumset A+B: the set of all sums formed by choosing one element from A and one from B. Both A and B must consist of nonnegative integers and contain at least two elements; either may be finite or infinite.

What does that help mathematicians do?

One concrete consequence is that a fixed finite pattern of at least two nonnegative integers cannot generate exactly the primes beyond some cutoff by taking copies shifted by nonnegative integers. Researchers can therefore rule out this repeated-pattern explanation even when overlaps and finitely many errors are allowed. The obstruction concerns complete coverage with no extra large numbers, not merely finding additive patterns within the primes.

Are there practical applications?

The immediate value is foundational: the result would establish that even eventual agreement with the primes is incompatible with a nontrivial sumset description. It distinguishes exact additive construction from the presence of individual sums or patterns. The supplied abstract reports no computational method or practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The additive indecomposability of the primes

September 24, 2026 80 pages

We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as A+BA+B, where A and B are sets of nonnegative integers with at least two elements each.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/013.md.

Ostmann’s inverse Goldbach conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves Ostmann's inverse Goldbach conjecture. For any two sets A,BA,B of nonnegative integers, each containing at least two elements, the symmetric difference between their sumset A+BA+B and the set of primes is infinite. Thus no set differing from the primes by only finitely many elements can be decomposed as such a sumset. The linked statements include the full result and its two-infinite-summands case.

Comparator links

Result Comparator statement
Ostmann's inverse Goldbach theorem and the infinite-summands case OstmannComplete.lean
Additive indecomposability of the primes up to finite changes OstmannPrimes.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.