On a sum of positive rational numbers whose product is 1

TL;DR

This paper proves that for any two integers m ≥ n ≥ 9, one can find n positive rational numbers with a product of 1 and a sum of m, expanding understanding of rational number configurations.

Contribution

It establishes the existence of specific rational number sets with prescribed sum and product constraints for a broad range of integers.

Findings

Existence of rational numbers with sum m and product 1 for m ≥ n ≥ 9

Construction method for such rational numbers

Generalization of sum-product relationships in rational numbers

Abstract

In this note, we prove that for every two positive integers $m \geq n \geq 9$, there exist $n$ positive rational numbers whose product is 1 and sum is $m$.