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arXiv · 2212.00918

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

Hongjian Li·Pingzhi Yuan·Hairong Bai
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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.

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Hongjian Li, Pingzhi Yuan, Hairong Bai. 2022-12-02. Positive rational number of the form $\varphi(km^{a})/\varphi(ln^{b})$. https://arxiv.org/abs/2212.00918

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