## Abstract

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 language | English |
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Pages (from-to) | 1401 - 1427 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 150 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Diophantine equations
- Erdos-Straus equation
- unit fractions

## ASJC Scopus subject areas

- Mathematics(all)