Ostmann's inverse Goldbach conjecture
Two subsets of are asymptotically equal if their symmetric difference is finite. Ostmann conjectured that the set of primes is asymptotically additively indecomposable. Is it true that there are no sets , each with at least two elements, such that differs from in only finitely many elements?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Additive number theory; sumsets and the primes
- Posed by
- Hans-Heinrich Ostmann (Additive Zahlentheorie, 1956, p. 13); restated by Elsholtz and Harper (2015, Conjecture 1.2)
- Year posed
- 1956
- Years open
- 70y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if have , then is infinite. After the Laffer-Mann reduction to two infinite summands, the proof combines residue restrictions at every prime, decorrelation from translated multiplicative characters, prime coverage and a positive statistic averaged over arrangements; it does not use Green and Harper's conditional inverse-sieve route. In particular, if contains every large prime it contains infinitely many composites. No quantitative version is claimed.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Ostmann's conjecture. lean/docs/013.md lists OstmannPrimes.json (OAI.Ostmann.main, module OAI.NumberTheory.Ostmann.Main) and OstmannComplete.json (OAI.Ostmann.inverseGoldbach and OAI.Ostmann.twoInfiniteSummandsImpossible, module OAI.NumberTheory.Ostmann.Complete). Both solution files exist at the pinned commit; neither challenge is in the formalization catalogue formalization.yaml. The statement was read here: for all sets A, B of natural numbers with at least two elements each, the symmetric difference of A+B and the primes is infinite. That is exactly the conjecture. Not rebuilt here.