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Claim. For integers $0<a<b$ let $N(a,b)$ be the least $n$ for which $a/b=1/x_1+\cdots+1/x_n$ with integers $0<x_1<\cdots<x_n$, and let $N(b)=\max_{1\le a<b}N(a,b)$, the function of Problem 304. Erdős's paper of 1950 proves two theorems about it. Theorem 1 (p. 195): there is a constant $c_1$ with

$$N(a,b)<c_1\,\frac{\log b}{\log\log b}$$

for every $0<a<b$; the proof gives $c_1=8$ for $b>4096$ (p. 202). Theorem 2 (p. 195): for every positive integer $b$,

$$N(b-1,b)>\log\log b-1 \qquad\text{and}\qquad \frac1{b-2}\sum_{a=1}^{b-2}N(a,b)>\tfrac12(\log\log b-1).$$

Together, $\log\log b\ll N(b)\ll\log b/\log\log b$, the bounds the site's commentary credits to the paper, and the average of $N(a,b)$ over $a$ is $\gg\log\log b$. Theorem 1 writes $a/b$ over $n!$ with $(n-1)!<b\le n!$ and uses that every integer below $n!$ is a sum of at most $n$ distinct divisors of $n!$. Theorem 2 rests on the paper's Theorem 5, that in any representation of $1$ by $n$ distinct unit fractions every denominator is below the $n$th Sylvester number, so a representation of $1$ containing $1/b$ with $n$ terms forces $\log\log b<n$. The statements, locators and proof sketches are paged at Theorem 1 and Theorem 2.

Covers. The bounds $\log\log b\ll N(b)\ll\log b/\log\log b$ and the average bound only. Not covered: the question whether $N(b)\ll\log\log b$, which the paper states as probable (p. 195) and which the OpenAI release's accepted claim answers. The upper bound was improved to $\ll\sqrt{\log b}$ by Vose; the lower bound is also proved in Lean, by a different argument, on the Aristotle proof's page.

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

Acceptance. Refereed: P. Erdős, Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól, Mat. Lapok 1 (1950), 192--210 (in Hungarian, with an English summary on p. 210; MR 13,280b), a journal publication. The site's commentary credits the bounds to the paper, but the site labels the problem OPEN, so that credit is not reviewed evidence. The proofs are summarized, not verified, on the library pages.

Dating. The page is dated by the publication year; the journal volume gives no day, and the day in the page name is a placeholder.