A lower bound for the number of Egyptian fractions

Abstract

An Egyptian fraction is a sum of the form $1/n_1+\cdots +1/n_r$ where $n_1,\dots ,n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k\ge 4$ such that ${\ln }_{k}N\ge 3/2$ it holds $$\frac{\ln (|E_N|)}{\ln 2}\ge (2-\frac{3}{{\ln }_{k}N})\frac{N}{\ln N}\prod _{j=3}^k{\ln }_{j}N,$$ where ${\ln }_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition ${\ln }_{k}N\ge k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $|E_N|$ for large positive integers $N$, and we give a table of $|E_N|$ for $N$ up to $154$.