Egyptian Fractions with odd denominators

TL;DR

This paper investigates the number of solutions to the Egyptian fractions equation with odd denominators, providing new bounds that significantly improve previous estimates, especially for odd values of k.

Contribution

It establishes new double-exponential bounds on the number of solutions with odd denominators, advancing understanding of the solution count for these specific Egyptian fractions.

Findings

For odd k, the number of solutions S(k) is bounded between double-exponential functions.

The bounds are tighter than previous estimates, especially improving the lower bound.

The results reveal rapid growth in the number of solutions as k increases.

Abstract

The number of solutions of the diophantine equation $\sum_{i=1}^k \frac{1}{x_i}=1,$ in particular when the $x_i$ are distinct odd positive integers is investigated. The number of solutions $S(k)$ in this case is, for odd $k$: \[\exp \left( \exp \left( c_1\, \frac{k}{\log k}\right)\right) \leq S(k) \leq \exp \left( \exp \left(c_2\, k \right)\right) \] with some positive constants $c_1$ and $c_2$. This improves upon an earlier lower bound of $S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)$.