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Published December 19, 2025 | Version Published
Journal Article Open

A question of Erdős and Graham on Egyptian fractions

Creators

  • 1. ROR icon California Institute of Technology
  • 2. ROR icon Stanford University
  • 3. ROR icon Georgia Institute of Technology
  • 4. ROR icon University of Illinois at Chicago
  • 5. ROR icon University of California, San Diego

Abstract

Answering a question of Erdős 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(cx + o(1))n for an explicit constant cx increasing with x.

Copyright and License

© 2025 David Conlon, Jacob Fox, Xiaoyu He, Dhruv Mubayi, Huy Tuan Pham, Andrew Suk, and Jacques Verstraete. Licensed under a Creative Commons Attribution License (CC-BY)

Acknowledgement

We are grateful to the American Institute of Mathematics for hosting the SQuaREs project at which this work was initiated. We are also indebted to Zachary Chase and the user Lucia for bringing the MathOverflow post [11] to our attention.

Funding

Supported by NSF Awards DMS-2054452 and DMS-2348859.
Supported by NSF Awards DMS-2154129 and DMS-2452737.
Supported by NSF Award DMS-2103154.
Supported by NSF Awards DMS-1952767 and DMS-2153576.
Supported by a Clay Research Fellowship and a Stanford Science Fellowship.
Supported by an NSF CAREER Award and NSF Awards DMS-1952786 and DMS-2246847.
Supported by NSF Awards DMS-1800332 and DMS-2347832.

Files

2404.16016v2.pdf

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Additional details

Related works

Describes
Journal Article: https://discreteanalysisjournal.com/article/154329-a-question-of-erdos-and-graham-on-egyptian-fractions (URL)
Is new version of
Discussion Paper: arXiv:2404.16016 (arXiv)

Funding

National Science Foundation
DMS-2054452
National Science Foundation
DMS-2348859
National Science Foundation
DMS-2154129
National Science Foundation
DMS-2452737
National Science Foundation
DMS-2103154
National Science Foundation
DMS-1952767
National Science Foundation
DMS-2153576
Clay Mathematics Institute
Stanford University
National Science Foundation
DMS-1952786
National Science Foundation
DMS-2246847
National Science Foundation
DMS-1800332
National Science Foundation
DMS-2347832

Dates

Submitted
2024-04-25
Available
2025-12-19
Published

Caltech Custom Metadata

Caltech groups
Division of Physics, Mathematics and Astronomy (PMA) , Mathematics Department
Publication Status
Published