Positive rational number of the form $\varphi(km^{a})/\varphi(ln^{b})$

TL;DR

This paper characterizes when every positive rational number can be expressed as a ratio of Euler totient functions of specific polynomial forms, revealing conditions on parameters for such representations to exist and be unique.

Contribution

It establishes necessary and sufficient conditions for representing all positive rationals in the form (km^a)/(ln^b), including uniqueness when (a, b)>1.

Findings

Every positive rational can be expressed as (km^a)/(ln^b) iff (a, b)=1 or (a, b, k, l)=(2,2,1,1).

The representation is unique when (a, b)>1.

The paper provides a complete characterization of such representations.

Abstract

Let $k, l, a$ and $b$ be positive integers with $\max\{a, \, b\}\ge2$. In this paper, we show that every positive rational number can be written as the form $\varphi(km^{a})/\varphi(ln^{b})$, where $m, \, n\in\mathbb{N}$ if and only if $\gcd(a, \,b)=1$ or $(a, b, k, l)=(2,2, 1, 1)$. Moreover, if $\gcd(a, b)>1$, then the proper representation of such representation is unique.