Problem 206

Let $x>0$ be a real number. For any $n\geq 1$ let

R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x

be the maximal sum of $n$ distinct unit fractions which is $<x$ .

Is it true that, for almost all $x$ , for sufficiently large $n$ , we have

R_{n+1}(x)=R_n(x)+\frac{1}{m},

where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$ ? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)

Source: erdosproblems.com/206

Number theory
Unit fractions

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. Kovač's Theorem 1 (J. Number Theory 268 (2025), 39--48) shows that the set of $x>0$ whose best Egyptian underapproximations are eventually greedy has Lebesgue measure zero, so the assertion fails for almost every $x$ instead of holding for almost every $x$ . The site's label is DISPROVED (LEAN); the suffix is a catalog label explained under Formalization. The claim page Kovač 2024 records the result, its postings and the acceptance evidence (refereed publication and the curator's credit) from which the standing above derives.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 31.
Cu22: Curtiss, D. R., On Kellogg's Diophantine problem. Amer. Math. Monthly 29 (1922), no. 10, 380--387.
Er50b: Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól (On a Diophantine equation). Mat. Lapok 1 (1950), 192--210.
Na23: Nathanson, M. B., Underapproximation by Egyptian fractions. J. Number Theory 242 (2023), 208--234, doi:10.1016/j.jnt.2022.07.005; arXiv:2202.00191v2 (2022).
Ch23b: Chu, H. V., A threshold for the best two-term underapproximation by Egyptian fractions. Indag. Math. (N.S.) 35 (2024), no. 2, 350--375, doi:10.1016/j.indag.2024.01.006; arXiv:2306.12564v2 (2024).
Ko24b: Kovač, V., On eventually greedy best underapproximations by Egyptian fractions. J. Number Theory 268 (2025), 39--48, doi:10.1016/j.jnt.2024.09.004; arXiv:2406.07218v3 (2024).
KoTa26: Kovač, V. and Tang, Q., Eventually greedy best Egyptian underapproximations of rational numbers via optimal control. arXiv:2607.28387v2 (4 August 2026), 30 pages; preprint.

Problem 206

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