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

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

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 \((a, b, k, l)=(2,2, 1, 1)\). Mo...

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Veröffentlicht in:arXiv.org 2022-12
Hauptverfasser: Li, Hongjian, Yuan, Pingzhi, Bai, Hairong
Format: Artikel
Sprache:eng
Schlagworte:
Representations
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Zusammenfassung: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.
ISSN:2331-8422