On the periodicity of some Farhi arithmetical functions

Let \(k\in\mathbb{N}\). Let \(f(x)\in \Bbb{Z}[x]\) be any polynomial such that \(f(x)\) and \(f(x+1)f(x+2)... f(x+k)\) are coprime in \(\mathbb{Q}[x]\). We call $$g_{k,f}(n):=\frac{|f(n)f(n+1)... f(n+k)|}{\text{lcm}(f(n),f(n+1),...,f(n+k))}$$ a Farhi arithmetic function. In this paper, we prove that...

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Bibliographic Details
Published inarXiv.org
Main Authors Qing-Zhong Ji, Chun-Gang, Ji
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 03.05.2009
Subjects
Mathematical analysis
Mathematical functions
Periodic variations
Polynomials
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Summary:Let \(k\in\mathbb{N}\). Let \(f(x)\in \Bbb{Z}[x]\) be any polynomial such that \(f(x)\) and \(f(x+1)f(x+2)... f(x+k)\) are coprime in \(\mathbb{Q}[x]\). We call $$g_{k,f}(n):=\frac{|f(n)f(n+1)... f(n+k)|}{\text{lcm}(f(n),f(n+1),...,f(n+k))}$$ a Farhi arithmetic function. In this paper, we prove that \(g_{k,f}\) is periodic. This generalizes the previous results of Farhi and Kane, and Hong and Yang.
Bibliography:content type line 50
SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
ISSN:2331-8422