The number of solutions of the Erd\H{o}s-Straus Equation and sums of $k$ unit fractions

TL;DR

This paper establishes new upper bounds for the number of solutions to the Erdős-Straus equation, introduces an efficient algorithm for finding all solutions, and improves bounds on the number of representations of rational numbers as sums of unit fractions.

Contribution

It provides improved upper bounds, an efficient solution-finding algorithm, and enhanced lower bounds for representations of rationals as sums of three or more unit fractions.

Findings

At most O_{ε}(n^{3/5+ε}) solutions for fixed m and large n.

An algorithm with expected runtime O_{ε}(n^{ε}(n^3/m^2)^{1/5}) for finding all solutions.

Infinitely many primes p with many solutions to m/p as sum of three unit fractions.

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 $\mathcal{O}_{\epsilon}(n^{3/5+\epsilon})$ solutions of $\frac{m}{n}=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. 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 $\mathcal{O}_{\epsilon}\left(n^{\epsilon}\left(\frac{n^3}{m^2}\right)^{1/5}\right)$, for any $\epsilon >0$. 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 $m\in \mathbb{N}$ in every reduced residue class $e \bmod f$ there exist infinitely many primes $p$ such that the number of solutions of the equation $\frac{m}{p}=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ is $\gg_{f,m} \exp\left(\left(\frac{5\log 2}{12 \mathrm{lcm}(m,f)}+o_{f,m}(1)\right)\frac{\log p}{\log \log p}\right)$. Previously the best known lower bound of this type was of order $(\log p)^{0.549}$.