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

Christian Elsholtz, Stefan Planitzer

Research output: Contribution to journalArticlepeer-review


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
Pages (from-to)1401 - 1427
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number3
Publication statusPublished - 2020


  • Diophantine equations
  • Erdos-Straus equation
  • unit fractions

ASJC Scopus subject areas

  • Mathematics(all)


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