Short Egyptian fractions

Follow-up works

• Thomas F. Bloom and Wouter van Doorn, Optimal bounds for the length and number of unit fraction representations (hexagon:2610.00407v1, 2026-10-20). A human-written exposition and simplification of this paper's proofs of $N(b)\ll\log\log b$, $\log\log F(k)\gg k$ and $\log\log v(k)\gg k$, deriving Theorem 1 from the more elementary prescribed-denominator argument and discussing improved constants in Section 5. The same note is also hosted at erdosproblems.com.
• Thomas Bloom (@thomasfbloom), post on X (2026-10-14). A 12-post thread announcing the Bloom-van Doorn exposition, which describes the proof of the $O(\log\log b)$ bound as "short and elementary", combining sum-product theory and the weak Goldbach problem, and notes that the AI paper proved the bound twice, first in a long way and then "in a much shorter and simpler way".

Comments and discussion

• Thomas F. Bloom and Wouter van Doorn, Section 6 "Comparison to the AI-generated paper" (hexagon:2610.00407v1, October 2026). Says the later elementary argument for $v(k)$ also gives Theorem 1, so "the technical arguments of Sections 3-5 (occupying 15 pages, out of 33 total) are entirely redundant"; it also criticizes the exposition and attribution of novelty, and the authors state they had not read Sections 3-5 in detail.
• Thomas Bloom, Erdős Problem #304 (erdosproblems.com, last edited 2026-10-16). The problem page now refers to "the recent OpenAI proof that $N(b)\ll \log\log b$" and links the paper and the Bloom-van Doorn exposition.
• Thomas Bloom, Erdős Problem #293 (erdosproblems.com, last edited 2026-10-16). Credits the paper with the proof that $\log\log v(k)\gg k$ (the paper's prescribed-denominator corollary), after van Doorn and Tang's $\log v(k)\gg k^2$.
• Thomas Bloom, Erdős Problem #148 (erdosproblems.com, last edited 2026-10-16). Credits the paper with the proof that $\log\log F(k)\gg k$; the page cites Elsholtz-Planitzer for the best known upper bound on $F(k)$.

Formalization

• OpenAI math release, Lean scope notes for family 025 (OpenAI Math Release, 2026-10-06). Per these notes, the release's Lean development proves the $\Theta(\log\log b)$ order of $N(b)$, $\log\log F(k)=\Theta(k)$, and $e^{e^{k/600}}\le v(k)\le 1+k^{2^{k-1}}$ for large $k$, via the comparator challenges EgyptianFractions.lean and ShortEgyptianFractions.lean.
• Google DeepMind formal-conjectures, FormalConjectures/ErdosProblems/304.lean (formal-conjectures). The statement upper_bound ($N(b)=O(\log\log b)$) is tagged research solved and links to the release's Lean proof lean/OAI/NumberTheory/ShortEgyptian/Main.lean, while the local theorem body remains sorry; the change came in PR #6898 (merged 2026-10-11).
• thepriceisright/publications, Lower1950.lean (GitHub, commit 8f055eb; accessed 2026-10-21). A separate Aristotle-generated formalization of the classical lower bound for Erdős Problem 304, compiled against pinned formal-conjectures definitions; it does not cover the paper's new upper bound. [AI-generated per source]

Related prior literature mentioned by others

• Wouter van Doorn (with ChatGPT), Practical numbers and Egyptian fractions (GitHub, uploaded 2026-09-16). An earlier, mostly AI-written note with Lean formalization proving $N(b)\le 2c_0(\log\log b)^2$ and $v(k)\ge\exp(\exp(\sqrt{k/(2c_0)}))$, with $c_0=14/\log 2$, via practical numbers; Bloom-van Doorn cite it as the previous best bound. [AI-generated per source]
• Ernest S. Croot III, On a coloring conjecture about unit fractions (Ann. of Math. 157 (2003), 545-556). Bloom-van Doorn say the Fourier method in the paper's proof of Theorem 1 is closely related to Croot's work; the paper cites Croot's 1999 Mathematika paper but not this one.
• M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions (J. Number Theory 8 (1976), 157-168). Bloom-van Doorn note that Erdős and Graham said the bound $v(k)\gg k!$ should follow from this work.
• K. Soundararajan, Approximating 1 from below using n Egyptian fractions (arXiv:math/0502247, 2005-02-11). Cited by Bloom-van Doorn in Appendix A for the classical approximation bound used to obtain the complementary bounds for the three results.
• Thomas F. Bloom and Christian Elsholtz, Egyptian fractions (arXiv:2210.04496; Nieuw Arch. Wiskd. 2022). A survey of open problems and recent progress on unit fractions, cited by Bloom-van Doorn for background.

Digests and expositions

• heyprof, Short Egyptian fractions (VisuApp, undated; accessed 2026-10-21). A paper-specific interactive exposition comparing greedy and shortest expansions and illustrating the historical length bounds.

News and blogs

• Azmat, Short Egyptian Fractions: How OpenAI’s Proof Solved Erdős’s 76-Year-Old Conjecture and Established the Optimal Double-Logarithmic Bound (Binary Verse AI, 2026-10-16). Walks through the paper's dense-family construction and two corollaries and discusses the published Lean scope.