On sums of Egyptian fractions

Abstract

Let $n,d$, and $k$ be positive integers where $n$ and $d$ are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval $[kd,\infty)$ of positive integers into a family of finite sets $X$ for which the sum of the reciprocals of the elements in $X$ is $n/d$. Theorem 1. There is a partition of $[2kd,\infty)$ into an infinite family of infinite sets $Y$ for which the sum of the reciprocals of the elements in $Y$ is $n/d$. Our method is grounded in the Vital Identity, $1/z = 1/(z+1) + 1/z(z+1)$, which holds for every complex number $z \notin \{-1,0\}$, and which gives rise to an eponymous algorithm that serves as our tool. At the core of our Theorems 1 and 2 is the number theoretic function $\star: x\mapsto \star x := x(x+1)$ into whose properties this paper continues an investigation initiated in [7].