Problem 301

Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to

\frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k}

with distinct $a,b_1,\ldots,b_k\in A$ ?

Estimate $f(N)$ . In particular, is it true that $f(N)=(\tfrac{1}{2}+o(1))N$ ?

Source: erdosproblems.com/301

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Open: the site's label is OPEN (page last edited 16 January 2026; as of 2026-10-07), and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is open, claim none: the four claim pages, van Doorn's upper bound 25/28 (recorded from the site's commentary), Wang's upper bound 667/806, Della Pietra's lower bound above one half and Della Pietra's upper bound 15437/19344, are pending partial claims, none of which would settle the estimation question; the lower bound, an AI-assisted proof claim of July 2026 without acceptance evidence, would answer the particular question in the negative if correct. No proof, disproof or accepted resolution was found in the search whose scope the Current assessment records. The bounds supported by sources read here are $N/2\le f(N)\le(25/28+o(1))N$ : the lower bound from the interval $(N/2,N]$ (elementary, checked here) and the upper bound from van Doorn's argument in the site's commentary (elementary; its two counting facts checked here). 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), p. 37 (the site gives no page). Library home: Erdős and Graham 1980.
Wa26: Wang, Xinjun, A 667/806 Upper Bound for Erdős Problem #301 on Unit-Fraction-Free Sets. Unpublished manuscript dated 27 May 2026, posted on ResearchGate; not refereed, not cited by the site, described in the site's thread as AI-generated without a disclaimer. Theorem 1, p. 2. Library home: Wang 2026.
DP26: Della Pietra, D., A positive-density improvement for all-length unit-fraction-free sets. Draft of 30 July 2026 (10 pages) in the GitHub repository donalddellapietra/erdos-301-proof (head commit dated 30 July 2026 per the GitHub API); unrefereed; not filed in the library. Claim page 2026_07_30_della_pietra; the repository's README of the same date announces the upper bound of the claim page 2026_07_30_della_pietra_upper.
OEIS: Raza, H., Sequence A390394, The On-Line Encyclopedia of Integer Sequences (2025; entry last modified 10 August 2026, server time): $f(n)$ for $n\le60$ , computed by integer linear programming; read. - The site's commentary attributes the $25/28$ argument to Wouter van Doorn and the non-distinct remark to Stijn Cambie and Wouter van Doorn; neither has a written source beyond the site. The argument's claim page is 2025_09_16_van_doorn.

Problem 301

Status

Proof claims

Discussion