The number of solutions of the Erdos-Straus Equation and sums of k unit fractions

Christian Elsholtz, Stefan Planitzer

Research output: Contribution to journalArticle


We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most solutions of. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time, for any. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation is. Previously, the best known lower bound of this type was of order.

Original languageEnglish
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Publication statusPublished - 1 Jan 2019


  • Diophantine equations
  • Erdos-Straus equation
  • unit fractions

ASJC Scopus subject areas

  • Mathematics(all)

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