Positive rational number of the form φ(km^a)/φ(ln^b)

2022/12/02 by Hongjian Li, Pingzhi Yuan, Li, Hongjian +3
Mathematics · #11A25 #11D85 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.00918

openalex publication_date 2022/12/02 · openalex created_date 2022/12/17 · openalex updated_date 2026/07/28

Abstract

Let k, l, a and b be positive integers with max\a, b\≥2. In this paper, we show that every positive rational number can be written as the form φ(km^a)/φ(ln^b), where m, n∈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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