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Claim. Every rational $a/b$ with $0<a<b$ has a representation $a/b=1/x_1+\cdots+1/x_n$ in integers $0<x_1<\cdots<x_n$ with $n\ll\sqrt{\log b}$ terms; so $N(b)\ll\sqrt{\log b}$ for the function of Problem 304, superseding the bound $\log b/\log\log b$ of Erdős's 1950 paper. As the review Zbl 0558.10015 (Ming-Chit Liu) describes the paper, Vose first proves, by an argument of Erdős's 1950 paper, that there is an increasing sequence $N_k$ of positive integers such that every integer $m$ with $1<m<N_k$ is a sum of at most $O(\sqrt{\log N_{k-1}})$ distinct divisors of $N_k$, and then derives the bound from it. Van Doorn and Tang restate the theorem as their Lemma 2.2, with $N_K=4^{\alpha K^2}(p_1\cdots p_K)^2$ and denominators dividing, or $b$ times divisors of, $N_K$ (Section 3), and Liu and Sawhney restate it (arXiv:2404.07113v1, p. 3). The paper itself is not held; the statement is taken from the review and these restatements.

Covers. The upper bound $N(b)\ll\sqrt{\log b}$ only. Not covered: the question whether $N(b)\ll\log\log b$, which the OpenAI release's accepted claim answers and which implies this bound.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: M. D. Vose, Egyptian fractions, Bull. London Math. Soc. 17 (1985), no. 1, 21--24, doi:10.1112/blms/17.1.21, a journal publication. The site's commentary credits the bound to the paper, but the site labels the problem OPEN, so that credit is not reviewed evidence.

Dating. The page is dated by the issue, no. 1 of volume 17 (1985); the record gives no day, and the day in the page name is a placeholder.