A question of Erd\H{o}s and Graham on Egyptian fractions

David Conlon; Jacob Fox; Xiaoyu He; Dhruv Mubayi; Huy Tuan Pham; Andrew Suk; Jacques Verstraëte

A question of Erd\H{o}s and Graham on Egyptian fractions

Combinatorics 2025-12-19 v2 Number Theory

Authors: David Conlon , Jacob Fox , Xiaoyu He , Dhruv Mubayi , Huy Tuan Pham , Andrew Suk , Jacques Verstraëte

Abstract

Answering a question of Erd\H{o}s and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$ .

Keywords

Egyptian fractions Erdős-Graham problem integer partitions counting asymptotics

Cite

@article{arxiv.2404.16016,
  title  = {A question of Erd\H{o}s and Graham on Egyptian fractions},
  author = {David Conlon and Jacob Fox and Xiaoyu He and Dhruv Mubayi and Huy Tuan Pham and Andrew Suk and Jacques Verstraëte},
  journal= {arXiv preprint arXiv:2404.16016},
  year   = {2025}
}

Comments

Reformatted for Discrete Analysis

A question of Erd\H{o}s and Graham on Egyptian fractions

Abstract

Keywords

Cite

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