Sums of four and more unit fractions and approximate parametrizations

Christian Elsholtz*, Stefan Planitzer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We prove new upper bounds on the number of representations of rational numbers (Formula presented.) as a sum of four unit fractions, giving five different regions, depending on the size of (Formula presented.) in terms of (Formula presented.). In particular, we improve the most relevant cases, when (Formula presented.) is small, and when (Formula presented.) is close to (Formula presented.). The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the number of solutions. These ‘approximate parametrizations’ were the key point to enable computer programmes to filter through a large number of equations and inequalities. Furthermore, this result leads to new upper bounds for the number of representations of rational numbers as sums of more than four unit fractions.

Original languageEnglish
Pages (from-to)695-709
Number of pages15
JournalBulletin of the London Mathematical Society
Issue number3
Publication statusPublished - Jun 2021


  • 11D68 (primary)
  • 11D72 (secondary)

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing


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