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COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

Part of: Multiplicative number theory Diophantine equations

Published online by Cambridge University Press:  15 May 2013

CHRISTIAN ELSHOLTZ  and
TERENCE TAO
CHRISTIAN ELSHOLTZ*
Affiliation:
Institut für Mathematik A, Steyrergasse 30/II, Technische Universität Graz, A-8010 Graz, Austria
TERENCE TAO
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA email [email protected]
*
✉[email protected]
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Abstract

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For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation

$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$
with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that
$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$
 These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Keywords

number of solutions of diophantine equationErdős–Straus conjecture

MSC classification

Secondary: 11D68: Rational numbers as sums of fractions 11N37: Asymptotic results on arithmetic functions 11D72: Equations in many variables 11N56: Rate of growth of arithmetic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 1 , February 2013 , pp. 50 - 105
Copyright
Copyright ©2012 Australian Mathematical Publishing Association Inc. 

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