VibeMathedMath problems solved with AI
Sign in

Erdős Problem #3: sets with divergent reciprocal sum contain arbitrarily long arithmetic progressions

Erdős problem #3 · erdosproblems.com/3

If A⊆NA\subseteq\mathbb N satisfies ∑n∈A1/n=∞\sum_{n\in A}1/n=\infty, must AA contain arbitrarily long arithmetic progressions? This extends Szemeredi's theorem to sparse sets, including the primes. The case of three-term progressions was settled by Bloom and Sisask (2020); for general kk the best bound before this work, rk(N)≪kNexp⁡(−(log⁡log⁡N)ck)r_k(N)\ll_kN\exp(-(\log\log N)^{c_k}) of Leng, Sah and Sawhney, falls short of the summability over dyadic intervals that the conjecture needs, where rk(N)r_k(N) is the largest size of a subset of {1,…,N}\{1,\ldots,N\} with no kk-term progression. Does every set of positive integers with divergent reciprocal sum contain kk-term arithmetic progressions for every kk?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Additive combinatorics
Posed by
Paul Erdős
Year posed
1974
Years open
52y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
65 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 claims, for each fixed k≥3k\ge3, rk(N)≤CkNexp⁡(−ck(log⁡N)εk)r_k(N)\le C_kN\exp(-c_k(\log N)^{\varepsilon_k}) with unspecified positive constants depending on kk; equivalently a density-α\alpha subset of [N][N] has a kk-term progression once log⁡N≥Ak(2+log⁡(1/α))Ak\log N\ge A_k(2+\log(1/\alpha))^{A_k}. Summing over dyadic blocks gives Erdos's conjecture, and even the stronger criterion with weights (log⁡(2+a))B/a(\log(2+a))^B/a, and recovers Green-Tao for dense subsets of the primes. The method is a density increment on triangular polynomial cells with controlled precision. No improvement of the three-term exponent is claimed, the exponents εk\varepsilon_k are not optimized, and it does not give an asymptotic formula for rk(N)r_k(N) (Erdos Problem #142).

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Corollary 1.2 was read against the posed problem and states it exactly; it follows in a few lines from Theorem 1.1 by summing over dyadic intervals. Lean-checked on the release's own Comparator challenge ErdosReciprocal (OAI.Erdos3.manuscriptReciprocalProgressionTheorem) together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here. The quantitative density bound is outside the formal statement.

Sources

Changelog1 change

Discussion