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

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Fingerprint

### Keywords

- Diophantine equations
- Erdos-Straus equation
- unit fractions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The number of solutions of the Erdos-Straus Equation and sums of k unit fractions.** / Elsholtz, Christian; Planitzer, Stefan.

Research output: Contribution to journal › Article › Research › peer-review

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TY - JOUR

T1 - The number of solutions of the Erdos-Straus Equation and sums of k unit fractions

AU - Elsholtz, Christian

AU - Planitzer, Stefan

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Diophantine equations

KW - Erdos-Straus equation

KW - unit fractions

UR - http://www.scopus.com/inward/record.url?scp=85060882292&partnerID=8YFLogxK

U2 - 10.1017/prm.2018.137

DO - 10.1017/prm.2018.137

M3 - Article

JO - Proceedings / Royal Society of Edinburgh / A

JF - Proceedings / Royal Society of Edinburgh / A

SN - 0308-2105

ER -