<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.1in}
\setlength{\textheight}{8.4in}

\newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}}

\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\begin{center}
\vskip 1cm{\LARGE\bf 
A Note on a Theorem of Rotkiewicz
}
\vskip 1cm
\large
Gombodorj Bayarmagnai \\
Department of Mathematics \\
National University of Mongolia\\
Baga Toirog \\
Ulaanbaatar 14200\\
Mongolia\\
\href{mailto:bayarmagnai@smcs.num.edu.mn}{\tt bayarmagnai@smcs.num.edu.mn}\\
\end{center}

\vskip .2 in


\begin{abstract}
In 1961, Rotkiewicz presented a generalisation of the well-known fact  that $n$ divides 
$\varphi(a^n-1)$ for all positive integers $n$ and $a&gt;1$, where $ \varphi$ 
is Euler's 
totient function.  In this note, we extend his result to values of cyclotomic polynomials.
\end{abstract}

\section{Introduction}
Let $ \varphi$ be the Euler's totient function.  It is well known that $n \mid \varphi( a^n-1)$ for all positive integers $n$ 
and $a &gt; 1$ (see, e.g., Gunderson \cite{Gud}). Let $ \Phi_k$ be the homogeneous cyclotomic polynomial of order $k$, 
and let $d(n)$ be the number of divisors of $n$. Rotkiewicz \cite{Ro} generalized the above result as follows:
$$
		n^{\frac{d(n)}{2}}\ \bigl|\ \varphi(\Phi_1(a^n,b^n))
$$
for all positive integers $a,b\ (a&gt;b)$ and $n$. In this note we extend this result to values of cyclotomic polynomials.
\begin{theorem}
		Let $n$ and $k$ be relatively prime positive integers. For all positive integers $a,b$ $(a&gt;b)$ 
		we have $$   k^{ \alpha } n^{ \frac{ d (n) }{2} }  \mid  \varphi( \Phi_k (a^n, b^n) ), $$ where	 
\begin{displaymath}
\alpha= \begin{cases} 
	d(n)-1, &amp; \text{if $a=2b$ and $ke=6$ for some $e\mid n$}; \\
	d(n), &amp; \text{otherwise.}
\end{cases}
\end{displaymath}
\end{theorem} 
Note that the case of $k =2$ was discussed in Rotkiewicz  \cite[Theorem 2]{Ro}.

Fix positive integers $a, b$  $(a&gt;b)$ and $k$, and define a sequence $( V_n^{ (k) } )_{n \geq 1}$ by setting 
$V_n^{(k)} = \Phi_k ( a^n, b^n )$. Since $\Phi_k$ is homogeneous, we may assume without loss of generality 
that $a$ and $b$ are relatively prime.

For convenience, we recall the notion of arithmetic primitive factor introduced in Birkhoff-Vandiver \cite{BV} in 
the following way.  A prime of $V_n^{ (k)}$ is called a primitive prime factor of the term if it does not divide any 
$V_m^{(k)}$ for proper divisors $m$ of $n$. We consider the arithmetic primitive factor of $V_n^{(k)}$ given 
by the product
$$
		P_n^{(k)}=\prod_pp^{v_p(V_n^{(k)})},
$$
where $p$ runs through all primitive prime factors of the term. Here, $v_p(n)$ denotes the exponent 
of $p$ in the decomposition of $n$. If $n$ and $k$ are relatively prime then it follows from the identity
\begin{align}\label{identity}
			\Phi_k(a^n,b^n)=\prod_{e\mid n} \Phi_{ke}(a,b)
\end{align}
that $P_n^{(k)}$ divides $\Phi_{kn}(a,b)$. 
\section{Proof}  
Let $n$ be an integer relatively prime to a prime $p$, and let ${\rm ord}_p(n)$  be the order of $n$  modulo $p$. 
We now state the following useful lemma.
\begin{lemma}\label{lemma}
		 Let $p$ be a prime not dividing $b$. Then
\begin{align*}
	&amp;(a)\	\    v_p(\Phi_k(a,b))\ne 0\text{ if and only if } k=p^{v_p(k)}{\rm ord}_p(ab^{-1}),
\\
	&amp;(b)\ \ 	\text{ if } v_p(k)\ne0 \text{ then } v_p(\Phi_k(a,b))\leq1\ (except\ k = p = 2).
\end{align*}
\end{lemma}
\begin{proof}
 See  Roitman \cite{Roi}.
 \end{proof}
 \begin{proof}[Proof of Theorem]
Let $d$ be a divisor of $n$. The identity (\ref{identity}) implies that every primitive prime of $ V^{(1)}_{kd} $ is a 
factor of $P_d^{(k)}$. Hence, by Zsigmondy's theorem, $P_d^{(k)} \ne 1$ if 
\begin{align}\label{condition}
	 (kd,a,b)\neq (6, 2, 1).
\end{align} 
Under the condition (\ref{condition}), we claim that $P_d^{(k)}$ has a prime factor not dividing $kd$. Suppose that $p$ is a 
prime of $kd$ dividing $ \Phi_{kd} (a, b)$. Then Lemma~\ref{lemma}$(a)$ implies that $ kd/p^{v_p(kd)} &lt;  p$ and so $p$ is 
the largest prime of $kd$. Thus, by Lemma~\ref{lemma}$(b)$, $p$ is  the greatest common divisor of $kd$ and $\Phi_{kd}
(a, b)$. Hence, if the claim is not true, then it follows that $P_d^{ (k) }$ equals the largest prime of $kd$. Moreover, it also 
equals the primitive factor $P^{(1)}_{kd}$. But this contradicts to the fact that $P_n^{ (1) }$ is prime to $p$ if the largest prime 
$p$ of $n$ is a factor of $V_n^{ (1) }$ (see Birkhoff-Vandiver \cite[Theorem 4 ]{BV}).

Next we have that the primitive factors $P_d^{(k)}$ are pairwise relatively prime. Indeed, if $p$ is a factor of $P_{d_1}^{(k)}$ 
and $P_{d_2}^{ (k)}$ then we may apply Lemma~\ref{lemma}$(a)$  to conclude that $d_1/d_2$ is a power of $p$. Hence, 
$p$ is not a primitive factor of one of $V_{ d_1}^{ (k) }$ and $V_{ d_2}^{ (k) }$. This is a contradiction.

Assume that (\ref{condition}) holds for each factor $d$ of $n$. Let $q$ be a prime factor of $ P_d^{(k)}$ not dividing $kd$. 
Then it follows from Lemma~\ref{lemma}$(a)$ that $kd\mid q-1$. Hence we obtain
\begin{align}\label{case}
		  k^2n\ \bigl|\ \varphi\bigl(P_{d}^{(k)})\varphi(P_{\frac{n}{d}}^{(k)}\bigl)
\end{align}
for each $d$ such that $n\ne d^2$.  Thus, it is now clear that the factor $\prod_{ d \mid n} \varphi(P_d^{(k)})$
of $\varphi(V_n^{(k)})$ is divisible by $k^{d(n)}n^{\frac{d(n)}{2}}$.

It remains to consider only the case  $(kd,a,b) = (6, 2, 1)$ with $d\mid n$. In this case  we have  
\begin{displaymath}
P^{(k)}_{\frac{6}{k}}=
\begin{cases}
1, &amp; \text{ if  $k$ is 1 or 2; }
\\
3, &amp; \text{ otherwise. }
\end{cases}
\end{displaymath}
Thus, (\ref{case}) implies that $kn \bigl|\ \varphi\bigl(P_{\frac{6}{k}}^{(k)})\varphi(P_{\frac{nk}{6}}^{(k)}\bigl)$ for $k=3, 6$.
When $k=2$, we combine (\ref{case}) with the fact that $2^3+1\mid V_n^{(2)}$. If $k=1$ then $P_{3}^{(1)}=7$ and so
$$n^2 \bigl|\ \varphi(P_{3}^{(1)}P_{\frac{n}{3}}^{(1)})\varphi(P_{\frac{n}{6}}^{(1)}\bigl)$$ as in the previous case.
This completes the proof.
\end{proof}

\section {Acknowledgment}\label{acknowledge}
The author would like to thank the referee for carefully reading the paper and for some suggestions.

\begin{thebibliography}{5}
\bibitem{BV} G. D. Birkhoff and H. S. Vandiver,  On the integral divisors of $a^n-b^n$, {\it Ann. of Math.} \textbf{5}  (1904), 173--180.

\bibitem{Gud} N. G. Gunderson, Some theorems on the Euler $\phi$-function,  {\it Bull. Amer. Math. Soc.}
\textbf{49} (1943), 278--280.

\bibitem{Ro} A. Rotkiewicz,  On the numbers $\varphi(a^n\pm b^n)$, {\it Proc. Amer. Math. Soc.} \textbf{12} (1961), 419--421.

\bibitem{Roi} M. Roitman, On Zsigmondy primes, {\it Proc. Amer. Math. Soc.} \textbf{125} (1997), 1913--1919.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11A25; Secondary 11B83.

\noindent \emph{Keywords: }  
Euler's totient function, primitive factor, cyclotomic polynomial.

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received November 25 2014;
revised versions received  February 4 2015; February 13 2015; 
February 14 2015.
Published in {\it Journal of Integer Sequences}, February 14 2015.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

                                                                                

</pre></body></html>