On Egyptian fractions of length 3

Abstract

Let $a,n$ be positive integers that are relatively prime. We say that $a/n$ can be represented as an Egyptian fraction of length $k$ if there exist positive integers $m_1, \ldots, m_k$ such that $\frac{a}{n} = \frac{1}{m_1}+ \cdots + \frac{1}{m_k}$. Let $A_k(n)$ be the number of solutions $a$ to this equation. In this article, we give a formula for $A_2(p)$ and a parametrization for Egyptian fractions of length $3$, which allows us to give bounds to $A_3(n)$, to $f_a(n) = \#\{(m_1,m_2,m_3) : \frac{a}{n} = \frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$, and finally to $F(n) = \#\{(a,m_1,m_2,m_3) : \frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$.