# Counting Egyptian fractions

## TL;DR

This paper investigates the growth of the number of Egyptian fractions with denominators up to N, providing bounds that involve iterated logarithms, thus advancing understanding of their combinatorial complexity.

## Contribution

It establishes new upper and lower bounds for the count of Egyptian fractions with denominators up to N, involving iterated logarithmic functions.

## Key findings

- Lower bound: (N / log N) times product of iterated logs
- Upper bound: approximately 0.421 times N
- Bounds hold for large N and fixed k ≥ 3

## Abstract

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.