Problem 295

Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with

1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.

Is it true that

\lim_{N\to \infty} k(N)-(e-1)N=\infty?

Source: erdosproblems.com/295

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Open: the site's label is OPEN. The Erdős–Straus solution of 1971 proves $(e-1)N-c_2<k(N)<(e-1)N+c_1N/\log N$ and says the divergence "seems to us certain" but is unproved; the 1980 monograph repeats both. No proof or disproof of the divergence was found in the search whose scope the Current assessment records; the one proof claim on the site is partial, AI-assisted and unreviewed. 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. 35.
ErSt71b: Ruderman, H. D. (proposer), Erdős, P. and Straus, E. G. (solvers), Problem E2232, Representation of 1 by Egyptian fractions. Amer. Math. Monthly 78 (1971), no. 3, 302--303, doi:10.2307/2317539.
Er50c: Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210; Theorem 1, p. 195, the input of the upper bound.
Cr01: Croot, E. S., III, On unit fractions with denominators in short intervals. arXiv:math/9904181 (1999); Acta Arith. 99 (2001), no. 2, 99--114, doi:10.4064/aa99-2-1. Context.
Ma99: Martin, G., Dense Egyptian fractions. arXiv:math/9804045 (1998); Trans. Amer. Math. Soc. 351 (1999), no. 9, 3641--3657, doi:10.1090/S0002-9947-99-02327-2. Context. - OEIS A192881, linked by the site.

Problem 295

Status

Proof claims

Discussion