Problem 282

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$ : choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$ . If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$ .

Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?

Source: erdosproblems.com/282

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Open on the site: the label is OPEN (no last-edited date shown;), and the site marks the problem as not resolvable by a finite computation. The frontmatter standing open, claim none, rests on no claim page: the site's proof-claim tab is empty, and no manuscript located claims either question. No proof, disproof or accepted resolution of the odd-denominator question, and no classification of the terminating pairs $(x,A)$ , was found in the search whose scope the Current assessment records. The published sources cited state the odd-denominator question as open (Pihko 2001 and 2010, Louwsma and Martino 2023, Koizumi 2025); the results recorded are existence criteria, which say which rationals have a representation at all, and termination for prescribed finite numbers of steps, none of which decides termination in general. This is a bounded negative finding, not a certificate of openness.

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), pp. 30--31 (the site cites p. 30). Library home: Erdős and Graham 1980.
Gr64b: Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) 14 (1964), 193--207, DOI 10.1112/plms/s3-14.2.193. The arithmetic-progression criterion, section 4(1), p. 206; the Stewart--Breusch existence theorem recalled on p. 193. Library home: Graham 1964.
We65: Webb, W. A., Sums of rational numbers. Canad. J. Math. 17 (1965), 1019--1024, DOI 10.4153/cjm-1965-096-3. The Breusch--Stewart theorem recalled as background, p. 1019; Theorem 2, pp. 1020--1023. Library home: Webb 1965.
MaSh21: Martin, G. and Shi, Y., An algorithm for Egyptian fraction representations with restricted denominators. Involve 18 (2025), 1--23; arXiv:2107.05076v1 (11 July 2021). Theorem 2.2 and the procedure of Section 3. Library home: Martin and Shi 2021.
LoMa25: Louwsma, J. and Martino, J., Rational numbers with two-term odd greedy expansion. Integers 25 (2025), A46, 7 pp., DOI 10.5281/zenodo.15536292. Theorem 3, pp. 4--5; Corollary 5, pp. 5--6. Library home: Louwsma and Martino 2025.
Gu04: Guy, R. K., Unsolved problems in number theory. Third edition, Springer (2004), Section D11, Egyptian fractions. Library home: Guy 2004.
Gr64c: Graham, R. L., On finite sums of reciprocals of distinct $n$ th powers. Pacific J. Math. 14 (1964), no. 1, 85--92, DOI 10.2140/pjm.1964.14.85. Theorem 4 and Corollary 1, p. 91. Library home: Graham 1964.
Pi01: Pihko, J., Remarks on the "greedy odd" Egyptian fraction algorithm. Fibonacci Quart. 39 (2001), no. 3, 221--227, DOI 10.1080/00150517.2001.12428725. Open Problem 1.1, p. 221; Theorem 2.3, p. 223. Library home: Pihko 2001.
Pi10: Pihko, J., Remarks on the "greedy odd" Egyptian fraction algorithm II. Fibonacci Quart. 48 (2010), no. 3, 202--208, DOI 10.1080/00150517.2010.12428097 (Crossref record accessed). The open-problem sentence, p. 202; Corollary 3.6, p. 206. Library home: Pihko 2010.
LoMa23: Louwsma, J. and Martino, J., Rational numbers with odd greedy expansion of fixed length. arXiv:2309.07280v1 (13 September 2023), 21 pp. Cited for its abstract. Library home: Louwsma and Martino 2023.
Ko25: Koizumi, J., Irrationality of the reciprocal sum of doubly exponential sequences. arXiv:2504.05933v1 (8 April 2025), 14 pp.; published as Integers 26 (2026), paper A28 (17 pp., which renumbers the results; the labels on this page are the preprint's). The odd-greedy passage, p. 7 of the preprint (journal p. 8). Library home: Koizumi 2025.
Stei58: The monograph's [Stei (58)], its source for Stein's question; its bibliography (printed p. 122) gives it as S. K. Stein, personal communication, so there is no published statement by Stein to cite.

Problem 282

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Proof claims

Discussion