Exceptions to the Erd\H os--Straus--Schinzel conjecture

TL;DR

This paper investigates exceptions to the Erdős–Straus–Schinzel conjecture, showing that for large enough m, there exist fractions m/n that cannot be expressed as a sum of three unit fractions, and provides explicit bounds and numerical evidence.

Contribution

The paper proves that if the bound n_m exists for all m, it must be extremely large, and provides explicit bounds and numerical data supporting the existence of exceptions.

Findings

Existence of large n where m/n is not a sum of 3 unit fractions.

Explicit bounds for m where exceptions occur.

Numerical evidence supporting theoretical results.

Abstract

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions.