Counting the number of solutions to the Erdos-Straus equation on unit fractions

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 \emph{Erd\H{o}s-Straus conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this conjecture, it suffices without loss of generality to consider the case when $n$ is a prime $p$. In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$ asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result establishes the asymptotic upper and lower bounds $$ N \log^2 N \ll \sum_{p \leq N} f(p) \ll N \log^2 N \log \log N.$$ In particular, from this bound and the prime number theorem we have $f(p) = O(\log^3 p \log \log p)$ for a subset of primes of density arbitrarily close to 1; thus a typical prime has a relatively small number of solutions to the Erd\H{o}s-Straus Diophantine equation. We also establish some related results on $f$ and related quantities, for instance establishing the bound $f(p) \ll p^{3/5} + O(\frac{1}{\log\log p})}$ for all primes $p$.