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\begin{center}
\vskip 1cm{\LARGE\bf 
Fermat Numbers in Multinomial Coefficients
}
\vskip 1cm
\large
Shane Chern \\
Department of Mathematics\\
Zhejiang University\\
Hangzhou, 310027\\
China\\
\href{mailto:chenxiaohang92@gmail.com}{\tt chenxiaohang92@gmail.com}\\
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\begin{abstract}
In 2001 Luca proved that no Fermat number can be a nontrivial
binomial coefficient. We extend this result to multinomial coefficients.
\end{abstract}

\section{Introduction}\label{Intro}

Let $F_m=2^{2^m}+1$ be the $m^{th}$ Fermat number for any nonnegative integer $m$. Several authors studied the Diophantine equation
\begin{equation}\label{eq:1.1}
\binom{n}{k}=2^{2^m}+1=F_m,
\end{equation}
where $n\geq 2k\geq 2$, and $m\geq 0$. We refer to the 
articles \cite{Hewgill,Krishna,Luca2,Luca,Radovici} for further details. In 2001, Luca \cite{Luca} completely solved Eq.~\eqref{eq:1.1} 
and proved that it has only the trivial solutions $k=1, n-1$ and $n=F_m$. 
The proof is mainly based on a congruence given by Lucas \cite{Lucas}. For more about Fermat numbers, see \cite{Krizek}.

For a positive integer $t$, let $n,k_1,\ldots,k_t$ be nonnegative integers, and define the $t$-order multinomial coefficient as follows:
$$\binom{n}{k_1,\ldots,k_t}=\frac{n(n-1)\cdots(n-k_1-\cdots-k_t+1)}{k_1!\cdots k_t!},$$
with $\sum_{i=1}^t k_i&lt; n+1$. In particular, $\binom{n}{0,\ldots,0}=1$. Note that for $t\geq 2$, if $\sum_{i=1}^t k_i= n$, then the
$t$-order multinomial coefficient equals 
a $(t-1)$-order multinomial coefficient
$$\binom{n}{k_1,\ldots,k_t}=\binom{n}{k_1,\ldots,k_{t-1}}.$$
There are many papers concerning the Diophantine equations related to multinomial coefficients. For example, Yang and Cai \cite{Yang} proved 
that the Diophantine equation
$$\binom{n}{k_1,\ldots,k_t}=x^l$$
has no positive integer solutions for $n,t\geq 3$, $l\geq 2$, and $\sum_{i=1}^t k_i= n$.

In this paper, we consider the Diophantine equation
\begin{equation}\label{eq:1.2}
\binom{n}{k_1,\ldots,k_t}=2^{2^m}+1=F_m,\quad\text{for $t\geq 2$, and $\sum_{i=1}^t k_i&lt; n$},
\end{equation}
and prove the following theorem.
\begin{theorem}\label{th:1.1}
The Diophantine equation~\eqref{eq:1.2} has no integer solutions $(m,n,k_1,\ldots,k_t)$ for nonnegative $m$ and positive $n,k_1,\ldots,k_t$.
\end{theorem}

\section{Two Lemmas}\label{Lemmas}

To prove Theorem \ref{th:1.1}, we need the following two lemmas.

\begin{lemma}[Euler \cite{Euler}]\label{lm:2.1}
Any prime factor $p$ of the Fermat number $F_m$ satisfies
$$p\equiv 1 \pmod{2^{m+1}}.$$
\end{lemma}

\begin{lemma}[Luca \cite{Luca}]\label{lm:2.2}
If $F_m=s\binom{n}{k}$, with $m\geq 5$, $s\geq 1$, and $1\leq k \leq \frac{n}{2}$, we have the following two properties.
\begin{enumerate}[(i)]
\item Let $n=n'd$, where
$$A=\left\{ p:\text{\rm{prime }}p\mid n,\text{\rm{ and }}p\equiv 1 \text{\rm{ (mod $2^{m+1}$)}}\right\},$$
and
$$n'=\prod_{p\in A}p^{\alpha_p}.$$
Then $k=d&lt; 2^m$.

\item $k-i\mid n-i$ for any $i=0,\ldots,k-1$.
\end{enumerate}
\end{lemma}
\begin{remark}
Lemma \ref{lm:2.2} is summarized from Luca's proof \cite{Luca} of
Diophantine equation (\ref{eq:1.1}). Although Luca only proved the case
$s=1$, he indicated that the result also holds for all positive integers $s$.
\end{remark}

\section{Proof of Theorem \ref{th:1.1}}\label{Proof}

The first five Fermat numbers are primes, which cannot be a multinomial
coefficient in Eq.~\eqref{eq:1.2}. Therefore, we only need to consider
$m\geq5$.

Moreover, for any multinomial coefficient $\binom{n}{k_1,\ldots,k_t}$ with $t&gt;0$, $k_1, \ldots, k_t\geq 1$, and $\sum_{i=1}^t k_i&lt; n$, there exists a multinomial coefficient
$$\binom{n}{k'_1,\ldots,k'_t}=\binom{n}{k_1,\ldots,k_t},$$
such that $1\leq k'_1,\ldots,k'_t\leq \frac{n}{2}$.

Hence, Eq.~\eqref{eq:1.2} becomes
\begin{equation}\label{eq:2.1}
F_m=\binom{n}{k_1,\ldots,k_t},\quad\text{for $m\ge 5$, $1\leq k_i\leq\frac{n}{2}$, and $\sum_{i=1}^t k_i&lt; n$.}
\end{equation}

Let $n=n'd$, where
$$A=\left\{ p:\text{prime }p\mid n,\text{ and }p\equiv 1 \text{ (mod $2^{m+1}$)}\right\},$$
and
$$n'=\prod_{p\in A}p^{\alpha_p}.$$
For any $i=1,\ldots,t$, we have
\begin{equation}\label{ki}
F_m=\binom{n}{k_1,\ldots,k_t}=\binom{n}{k_i}\binom{n-k_i}{k_1,\ldots,k_{i-1},k_{i+1},\ldots,k_t},
\end{equation}
where $\binom{n-k_i}{k_1,\ldots,k_{i-1},k_{i+1},\ldots,k_t}$ is a positive integer. By Lemma \ref{lm:2.2} (i), we have $k_i=d&lt;2^m$ for $i=1,\ldots,t$. 
Then Eq.~\eqref{eq:2.1} becomes
\begin{align}
F_m&amp;=\binom{n}{\underbrace{d,\ldots,d}_{t}},\quad \text{$n&gt;td$, and $t\geq 2$} \label{eq:2.2}\\
&amp;=\binom{n}{d}\binom{n-d}{d}\binom{n-2d}{\underbrace{d,\ldots,d}_{t-2}}. \label{eq:2.3}
\end{align}

Note that $d\geq 1$. We study Eq.~\eqref{eq:2.3} in the following three cases.

\bigskip

Case 1: $d&gt;2$. Since $n&gt;2d$ and $d\mid n$, we have $n\geq 3d$. Then,
$$d\leq \frac{n-d}{2}&lt;\frac{n}{2}.$$
In Eq.~\eqref{eq:2.3}, applying Lemma \ref{lm:2.2} (ii) to $\binom{n}{d}$ and $\binom{n-d}{d}$ respectively, and setting $i=1$, we have
$$d-1\mid n-1$$
and
$$d-1\mid n-d-1.$$
Thus, $d-1\mid d$, which is impossible.

\bigskip

Case 2: $d=2$. Let $n=2n'$. Then Eq.~\eqref{eq:2.2} becomes
$$F_m=\binom{2n'}{\underbrace{2,\ldots,2}_{t}}=n'(2n'-1)(n'-1)(2n'-3)\binom{2n'-4}{\underbrace{2,\ldots,2}_{t-2}}.$$
Then $n'$ and $n'-1$ are both $F_m$'s factors. According to Lemma \ref{lm:2.1}, we obtain $n'\equiv n'-1\equiv 1$ (mod $2^{m+1}$), which is impossible.

\bigskip

Case 3: $d=1$. Eq.~\eqref{eq:2.2} becomes
$$F_m=\binom{n}{\underbrace{1,\ldots,1}_{t}}=n(n-1)\binom{n-2}{\underbrace{1,\ldots,1}_{t-2}}.$$
Then $n$ and $n-1$ are both $F_m$'s factors. According to Lemma \ref{lm:2.1}, we obtain $n\equiv n-1\equiv 1$ (mod $2^{m+1}$), which is also impossible.

\bigskip

This completes the proof of Theorem \ref{th:1.1}.

\begin{remark}
One can even find that the multinomial coefficient in Eq.~\eqref{eq:1.2} could not divide a Fermat number. Otherwise, assume that there exists a positive integer $s$ such that
$$F_m=s\binom{n}{k_1,\ldots,k_t}.$$
Note that in Eq.~\eqref{ki} we still have
$$\binom{n}{k_i} \mid F_m,$$
and in Eqs.~\eqref{eq:2.2} and \eqref{eq:2.3} similar results hold. Hence, we can get the proof in the same way.
\end{remark}

\section{Acknowledgments}\label{Ack}
I am indebted to Mr. Yong Zhang for providing relevant references and examining the whole proof, and  to Mr. Jiaxing Cui for giving detailed comments.

I am also indebted to the referee for his careful reading and helpful suggestions.

\begin{thebibliography}{9}

\bibitem{Euler}
L. Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus, {\it Acad. Sci. Petropol.} \textbf{6} (1738), 103--107.
Available at \url{http://eulerarchive.maa.org/pages/E026.html}.

\bibitem{Hewgill}
D. Hewgill, A relationship between Pascal's triangle and Fermat's numbers, {\it Fibonacci Quart.} \textbf{15} (1977), 183--184.

\bibitem{Krishna}
H. V. Krishna, On Mersenne and Fermat numbers, {\it Math. Student}
\textbf{39} (1971), 51--52.

\bibitem{Krizek}
M. K\v{r}\'i\v{z}ek, F. Luca, and L. Somer, \emph{17 Lectures on Fermat
Numbers: From Number Theory to Geometry}, Springer, 2001.

\bibitem{Luca2}
F. Luca, Pascal's triangle and constructible polygons, {\it Util. Math.} \textbf{58} (2000), 209--214.

\bibitem{Luca}
F. Luca, Fermat numbers in the Pascal triangle, {\it Divulg. Mat.} \textbf{9} (2001), 191--195.

\bibitem{Lucas}
\'E. Lucas, Th\'eorie des fonctions num\'eriques simplement p\'eriodiques, {\it Amer. J. Math.} \textbf{1} (1878), 184--196, 197--240, 289--321.

\bibitem{Radovici}
P. Radovici-M\u{a}rculescu, Diophantine equations without solutions (Romanian), {\it Gaz. Mat. Mat. Inform.} \textbf{1} (1980), 115--117.

\bibitem{Yang}
P. Yang and T. Cai, On the Diophantine equation $\binom{n}{k_1,\ldots,k_s}=x^l$, {\it Acta Arith.} \textbf{151} (2012), 7--9.

\end{thebibliography}





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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11D61; Secondary 11D72, 05A10.

\noindent \emph{Keywords: }
Fermat number, multinomial coefficient.

\bigskip
\hrule
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\noindent (Concerned with sequence
\seqnum{A000215}.)

\bigskip
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\vspace*{+.1in}
\noindent
Received  January 19 2014;
revised version received  February 11 2014.
Published in {\it Journal of Integer Sequences}, February 15 2014.

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