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abstract: 'For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ with $x,y,z$ positive integers. The *Erdős-Straus conjecture* asserts that $f(n) > 0$ for every $n {\geqslant}2$. 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 $$N \log^2 N \ll \sum_{p{\leqslant}N} f(p) \ll N \log^2 N \log \log N.$$ 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.'
address:
- 'Institut für Mathematik A, Steyrergasse 30/II, Technische Universität Graz, A-8010 Graz, Austria'
- 'Department of Mathematics, UCLA, Los Angeles CA 90095-1555'
author:
- Christian Elsholtz
- Terence Tao
title: 'Counting the number of solutions to the Erdős-Straus equation on unit fractions'
---

Introduction
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For any natural number $n \in {\mathbb{N}}= \{1,2,\ldots\}$, let $f(n)$ denote the number of solutions $(x,y,z) \in {\mathbb{N}}^3$ to the Diophantine equation $$\label{xyz}
\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$

(we do not assume $x,y,z$ to be distinct or in increasing order). Thus for 