Problem 284

Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<\cdots <n_k$ with

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

Is it true that

f(k)=(1+o(1))\frac{k}{e-1}?

Source: erdosproblems.com/284

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved, in the site's label. The status-defining source is Croot's Main Theorem (Acta Arith. 99 (2001), no. 2, 99--114; refereed): for every $N>1$ there is a representation $1=\sum1/x_i$ with $N<x_1<\cdots<x_k\le(e+o(1))N$ , which gives $f(k)=(1+o(1))k/(e-1)$ for the infinitely many $k$ that occur as term counts of these representations. The paper states the result "for infinitely many $k$ "; the site's commentary calls the problem essentially solved by Croot, and its PROVED label rests on that credit. The standing in the frontmatter derives from the accepted claim page Croot 1999.

Cr01: Croot, III, Ernest S., On unit fractions with denominators in short intervals. Acta Arith. 99 (2001), no. 2, 99--114, DOI 10.4064/aa99-2-1 (received 11 June 1999, revised 18 September 2000); arXiv:math/9904181v1 (30 April 1999, the only arXiv version, 19 pages). The Main Theorem is on p. 100 of the published version and p. 1 of the preprint. Library home: Croot 1999; result page Croot 1999, Main Theorem.
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), printed p. 33 (the page is supplied by the thread's comment and by Croot's reference [3]; the site's source key carries none). Library home: Erdős and Graham 1980.
Ma00: Martin, Greg, Denser Egyptian fractions. Acta Arith. 95 (2000), no. 3, 231--260. Croot's introduction says it gives no information about $x_1$ ; its Theorem 2 is compiled on Problem 285.

Problem 284

Status

Proof claims

Discussion