Problem 292

Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$ . Explore $A$ . In particular, does $A$ have density $1$ ?

Source: erdosproblems.com/292

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved, in the site's label, and the answer is yes: Martin's Theorem 4 (Acta Arith. 95 (2000), no. 3, 231--260; refereed) shows that $\mathcal L_1(r)$ has zero density for every positive rational $r$ , with counting function of exact order $x\log\log x/\log x$ ; at $r=1$ this is $|B\cap[1,x]|\asymp x\log\log x/\log x$ , so $A$ has density $1$ . This is the accepted claim Martin 2000, refereed and credited by the site's curator.

Ma00: Martin, Greg, Denser Egyptian fractions. Acta Arith. 95 (2000), no. 3, 231--260, DOI 10.4064/aa-95-3-231-260; arXiv:math/9811112v1 (18 November 1998, the only arXiv version, 26 pages; no file is held). Theorems 3 and 4, p. 3 of the preprint; proofs in Sections 6 and 7. Library home: Martin 2000.
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. Library home: Erdős and Graham 1980.
OEIS: Alekseyev, M., Sequence A092671, The On-Line Encyclopedia of Integer Sequences (2004; entry last modified 5 November 2025, server time): the elements of $A$ to $10000$ (b-file by J. E. Schoenfield), the observations that no prime power lies in $A$ and that multiples of elements greater than $1$ lie in $A$ , and a conjectured characterization verified to $2\cdot10^5$ ; accessed.

Problem 292

Status

Proof claims

Discussion