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

Part of: Diophantine equations

Published online by Cambridge University Press:  30 January 2019

Christian Elsholtz  and
Stefan Planitzer
Christian Elsholtz
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Kopernikusgasse 24/II, Graz8010, Austria (elsholtz@math.tugraz.at; planitzer@math.tugraz.at)
Stefan Planitzer
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Kopernikusgasse 24/II, Graz8010, Austria (elsholtz@math.tugraz.at; planitzer@math.tugraz.at)
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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 ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. 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 ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. 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 $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

Keywords

unit fractionsErdős-Straus equationDiophantine equations

MSC classification

Primary: 11D68: Rational numbers as sums of fractions
Secondary: 11D72: Equations in many variables
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1401 - 1427
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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