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 $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds $$ \frac{\ln(|E_N|)}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , $$ where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erd\H{o}s who established a similar bound but under the more stringent condition $\ln_k N\geq k$ 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 $154$.