A question of Erd\H{o}s and Graham on Egyptian fractions

TL;DR

This paper investigates the number of representations of fixed positive rational numbers as sums of distinct unit fractions with denominators up to n, revealing an exponential growth rate characterized by an explicit constant.

Contribution

It provides an explicit exponential growth rate for the number of Egyptian fraction representations of fixed rationals, answering a question posed by Erdős and Graham.

Findings

Number of representations grows as 2^{c_x n} for each fixed x.

The constant c_x increases with x.

The growth rate is explicitly characterized.

Abstract

Answering a question of Erd\H{o}s and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$.