Sums of four and more unit fractions and approximate parametrizations

Christian Elsholtz*, Stefan Planitzer

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

Abstract

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.

Originalspracheenglisch
Seiten (von - bis)695-709
Seitenumfang15
FachzeitschriftBulletin of the London Mathematical Society
Jahrgang53
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - Jun 2021

Schlagwörter

  • Diophantine equations
  • unit fractions

ASJC Scopus subject areas

  • !!Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

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