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Erdős Problem 304: are all fractions a/ba/b sums of O(log⁡log⁡b)O(\log\log b) distinct unit fractions?

Erdős problem #304 · erdosproblems.com/304

For integers 1≤a<b1\le a<b let N(a,b)N(a,b) be the least kk such that a/b=1/n1+⋯+1/nka/b=1/n_1+\cdots+1/n_k with integers 1<n1<⋯<nk1<n_1<\cdots<n_k, and let N(b)=max⁡1≤a<bN(a,b)N(b)=\max_{1\le a<b}N(a,b). Erdos (1950) proved log⁡log⁡b≪N(b)≪log⁡b/log⁡log⁡b\log\log b\ll N(b)\ll\log b/\log\log b, and Vose (1985) improved the upper bound to N(b)≪log⁡bN(b)\ll\sqrt{\log b}. Is it true that N(b)≪log⁡log⁡bN(b)\ll\log\log b?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Egyptian fractions; additive combinatorics
Posed by
Paul Erdős (1950, Matematikai Lapok); repeated in Erdős and Graham (1980)
Year posed
1950
Years open
76y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are absolute c1,c2>0c_1,c_2>0 and b0b_0 with c1log⁡log⁡b≤N(b)≤c2log⁡log⁡bc_1\log\log b\le N(b)\le c_2\log\log b for b≥b0b\ge b_0; the upper bound holds for every numerator. Corollaries: the number F(k)F(k) of kk-term expansions of 1 satisfies log⁡log⁡F(k)≍k\log\log F(k)\asymp k (Erdős-Graham asked for estimates), and the least integer v(k)≥2v(k)\ge2 missing from all kk-term expansions of 1 satisfies eek/600≤v(k)≤1+k2k−1e^{e^{k/600}}\le v(k)\le 1+k^{2^{k-1}} eventually, so log⁡log⁡v(k)≍k\log\log v(k)\asymp k, answering the growth question of Erdős Problem 293 up to constants. Not shown: the optimal constants, or control of denominator sizes (unlike Tenenbaum-Yokota).

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. 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 manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026.

Verification

No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Erdős Problem 304 as stated on erdosproblems.com. Lean (in lean/formalization.yaml): OAI.Problem337.main_double_log_order in OAI/NumberTheory/EgyptianFractions/Main.lean states constants c1,c2>0c_1,c_2>0 and b0b_0 with c1log⁡log⁡b≤max⁡1≤a<bN(a,b)≤c2log⁡log⁡bc_1\log\log b\le\max_{1\le a<b}N(a,b)\le c_2\log\log b for b≥b0b\ge b_0, with NN the least length of an expansion into strictly increasing denominators at least 2; a companion theorem shows that least length is attained. That is the headline. A second challenge, ShortEgyptianFractions (OAI.ShortEgyptian.main), states the same order. Not rebuilt here.

Sources

Changelog1 change

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