On sums of Egyptian fractions

TL;DR

This paper proves the existence of partitions of certain infinite intervals into sets whose reciprocals sum to a given rational number, using the Vital Identity and number theoretic functions, advancing the understanding of Egyptian fractions.

Contribution

It introduces new partitions of infinite intervals into sets with prescribed reciprocal sums, employing the Vital Identity and exploring properties of the function x↦x(x+1).

Findings

Constructed partitions of [kd,∞) with reciprocal sum n/d.

Constructed partitions of [2kd,∞) with reciprocal sum n/d.

Developed an algorithm based on the Vital Identity for 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].