On the representation of rational numbers via Euler's totient function

TL;DR

This paper explores how positive rational numbers can be expressed using Euler's totient function with specific algebraic forms, expanding understanding of number representations through totatives.

Contribution

It introduces new formulas showing that all positive rationals can be represented via Euler's totient function in specific algebraic forms, extending previous number theory results.

Findings

Every positive rational can be expressed as (m^{2})/(\u00f4(n^{2}))^{b}

Every positive rational can be expressed as (k(m^{2}-1))/(ln^{2})

Additional related results are discussed in the paper.

Abstract

Let $b>1$ be an odd positive integer and $k, l \in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $\varphi(m^{2})/(\varphi(n^{2}))^{b}$ and $\varphi(k(m^{2}-1))/\varphi(ln^{2})$, where $m, n\in \mathbb{N}$ and $\varphi$ is the Euler's totient function. At the end, some further results are discussed.