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\begin{center}
\vskip 1cm{\LARGE\bf Giuga Numbers and \\
\vskip .1in
the Arithmetic Derivative
}
\vskip 1cm
\large
Jos\'{e} Mar\'{i}a Grau \\
Departamento de Matematicas \\
Universidad de Oviedo\\ 
Avda.\ Calvo Sotelo, s/n \\
33007 Oviedo \\
Spain\\
\href{mailto:grau@uniovi.es}{\tt grau@uniovi.es} \\
\ \\
Antonio M. Oller-Marc\'{e}n\\
Centro Universitario de la Defensa\\ 
Ctra.\ de Huesca, s/n \\
50090 Zaragoza \\
Spain \\
\href{mailto:oller@unizar.es}{\tt oller@unizar.es}
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\begin{abstract}
We characterize Giuga numbers as solutions to the equation $n'=an+1$,
with $a \in \mathbb{N}$ and $n'$ being the arithmetic derivative.
Although this fact does not refute Lava's conjecture, it does suggest
doubts about its veracity.
\end{abstract}

\section{Introduction}
\subsection{The Arithmetic Derivative}

The arithmetic derivative was introduced by Barbeau \cite{BAR}
(see \seqnum{A003415} in Sloane's \emph{On-Line Encyclopedia of Integer
Sequences}). The derivative of an integer is defined to be the unique map sending
every prime to 1 and satisfying the Leibnitz rule, i.e., $p'=1$, for any
prime $p$, and $(nm)'=nm'+mn'$ for any $n,m\in \mathbb{N}$. This map makes
sense and is well-defined \cite{VIC}.

\begin{proposition} 
If $n=\prod_{i=1}^{k}p_i^{r_i}$ is the factorization of $n$ in prime
powers, then the only way to define $n'$ satisfying the desired properties is
$$n'=n\sum_{i=1}^{k}\frac{r_i}{p_i}.$$
\end{proposition}

\subsection{Giuga numbers} 

In \cite{BOR}, Giuga numbers were introduced in the following way motivated by previous work by Giuga \cite{GIU}.

\begin{definition}
A Giuga number is a composite number $n$ such that $p$ divides $\frac{n}{p}-1$ for
every prime divisor $p$ of $n$.
\end{definition}

It follows easily from the definition that every Giuga number is square-free.. There are several characterizations of Giuga numbers. We present the most important
in the following proposition (note that we will consider $\mathbb{N}=\{1,2,\dots\}$).

\begin{proposition}
Let $n$ be a composite integer. Then, the following are equivalent:
\begin{itemize}
\item [i)] $n$ is a Giuga number.
\item [ii)] Giuga \cite{GIU}: $\displaystyle{\sum_{p|n} \frac{1}{p}-\prod_{p|n} \frac{1}{p}
\in \mathbb{N}}$.
\item[iii)] Borwein et al. \cite{BOR}: $\displaystyle{\sum_{j=1}^{n-1}  j^{\phi(n)} \equiv -1}$ (mod
$n$), where $\phi$ is Euler's totient function.
\item [iv)] Agoh \cite{AGO}: $nB_{\phi(n)} \equiv -1$ (mod $n$), where $B$ is a Bernoulli
number.
\end{itemize}
\end{proposition}

To date, only thirteen Giuga numbers are known (see \seqnum{A007850} in 
Sloane's \emph{On-Line Encyclopedia of Integer Sequences}):
\begin{itemize}
\item With 3 factors:
$$\textbf{30} = 2 \cdot 3 \cdot 5.$$
\item With 4 factors:
\begin{align*}
\textbf{858}&=2 \cdot 3 \cdot 11 \cdot 13,\\
\textbf{1722}&=2 \cdot 3 \cdot 7 \cdot 41.
\end{align*}
\item With 5 factors:
$$\textbf{66198} = 2 \cdot 3 \cdot 11 \cdot 17 \cdot 59.$$
\item With 6 factors:
\begin{align*}
\textbf{2214408306}&=2 \cdot 3 \cdot 11 \cdot 23 \cdot 31 \cdot 47057,\\
\textbf{24423128562}&=2 \cdot 3 \cdot 7 \cdot 43 \cdot 3041 \cdot 4447.
\end{align*}
\item With 7 factors:
\begin{align*}
\textbf{432749205173838}&=2 \cdot 3  \cdot 7 \cdot 59 \cdot 163 \cdot 1381
\cdot 775807,\\
\textbf{14737133470010574}&=2 \cdot 3 \cdot 7 \cdot 71 \cdot 103 \cdot 67213
\cdot 713863,\\
\textbf{550843391309130318}&=2 \cdot 3 \cdot 7 \cdot 71 \cdot 103 \cdot
61559 \cdot 29133437.
\end{align*}
\item With 8 factors:
\begin{align*}
\textbf{244197000982499715087866346}=\ & 2 \cdot 3 \cdot 11 \cdot 23 \cdot
31 \cdot 47137 \cdot 28282147 \cdot \\ &3892535183,\\
\textbf{554079914617070801288578559178}=\ & 2 \cdot 3 \cdot 11 \cdot 23
\cdot 31 \cdot 47059 \cdot 2259696349 \cdot\\ & 110725121051,\\
\textbf{1910667181420507984555759916338506}=\ & 2 \cdot 3 \cdot 7 \cdot 43\cdots 1831 \cdot 138683 \cdot 2861051 \cdot \\ &1456230512169437.
\end{align*}
\end{itemize}

There are no other Giuga numbers with fewer than 8 prime factors. There is
another known Giuga number (found by Frederick Schneider in 2006) which has
10 prime factors, but it is not known if there is any Giuga number between
this and the previous ones. This largest known Giuga number is the following:

\begin{align*}
&\textbf{4200017949707747062038711509670656632404195753751630609228764416}\\
&\textbf{142557211582098432545190323474818}= 2 \cdot 3 \cdot 11 \cdot 23
\cdot 47059 \cdot 2217342227 \cdot\\ &1729101023519 \cdot
8491659218261819498490029296021 \cdot\\
&658254480569119734123541298976556403.
\end{align*}

Observe that all known Giuga numbers are even. If an odd Giuga number
exists, it must be the product of at least 14 primes. It is not even known
if there are infinitely many Giuga numbers.

In 2009, Paolo P. Lava, an active collaborator of the \emph{On-Line Encyclopedia of
Integer Sequences}, conjectured that Giuga numbers were exactly the
solutions of the differential equation $n'=n+1$, with $n'$ being the arithmetic
derivative of $n$. It is not a well-known conjecture (see the book by Lava \cite{LAVA} or comments about sequence \seqnum{A007850} in 
Sloane's \emph{On-Line Encyclopedia of Integer Sequences}) but it seems known enough to appear in wikipedia's article on Giuga numbers \cite{WIK}.

\section{Giuga numbers and the arithmetic derivative: bringing doubts on Lava's
conjecture}

It is surprising that Lava's conjecture has not been resolved. Certainly the thirteen known Giuga numbers satisfy $n'=n+1$. Nevertheless, we can observe that this is exclusively due to the fact that these thirteen known Giuga numbers satisfy $ \sum_{p|n} \frac{1}{p}-\prod_{p|n}
\frac{1}{p} =1$. Let us introduce now a novel characterization of Giuga numbers in terms of the arithmetic derivative.

To do so we first need the following technical lemma \cite[Corollary 2]{VIC}..

\begin{lemma} If $n'=an+1$, then $n$ is square-free.
\end{lemma}
\begin{proof}
If there is a prime $p$ such that $p^2$ divides $n$, then $p$ must divide $n'$. Since $n'=an+1$ this yields a contradiction.
\end{proof}

\begin{proposition} 
Let $n$ be an integer. Then $n$ is a Giuga number if and only if  $n'=an+1$ for some
$a\in\mathbb{N}$.
\end{proposition}
\begin{proof} 
If $n$ is a Giuga number then it is composite and square-free.
Thus, we can put
$n=p_1 p_2 \cdots p_k$ con $k>1$ and since $n$ is a Giuga number 
$$ \sum_{i=1}^k \frac{1}{p_i}-\prod_{i=1}^k\frac{1}{p_i}=\sum_{i=1}^k\frac{1}{p_i}
-\frac{1}{n}=a \in \mathbb{Z}.$$
Consequently, $\displaystyle{n\sum_{i=1}^k\frac{1}{p_i} -1=a n}$ and it follows by definition that $n'=a
n+1$.

Conversely, assume that $n'=an+1$ with $a\in \mathbb{N}$. Then $n$ is square-free, i.e., $\displaystyle{\prod_{p|n}\frac{1}{p}=\frac{1}{n}}$. Thus, $\displaystyle{an+1=n'= n\sum_{i=1}^k\frac{1}{p_i}}$ implies that $\displaystyle{\sum_{i=1}^k\frac{1}{p_i}-\frac{1}{n}=a\in \mathbb{N}}$, i.e., $n$ is a Giuga number, as claimed.
\end{proof}

This result shows that Lava's conjecture is as close (or far away) to
being refuted as the discovery of a Giuga number with

$$\sum_{i=1}^k\frac{1}{p_i}-\frac{1}{n}>1.$$ 

This is not likely to happen in the near future, since it is known that
such a number must have more than 59 prime factors \cite{survey}. In
any case it has been pointed out that Lava's conjecture is not
plausible and it could well be false.

\begin{thebibliography}{1}

\bibitem{AGO}
T. Agoh, On {G}iuga's conjecture, {\em Manuscripta Math.} {\bf 87}
(1995), 501--510.

\bibitem{LAVA}
G. Balzarotti and P. Lava, {\it 103 Curiosit\`a Matematiche}, Hoepli,
Milan, 2010.

\bibitem{BAR}
E. J. Barbeau, Remarks on an arithmetic derivative, {\em Canad. Math.
Bull.} {\bf 4} (1961), 117--122.

\bibitem{BOR}
D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn, Giuga's
conjecture on primality, {\em Amer. Math. Monthly} {\bf 103} (1996),
40--50.

\bibitem{survey}
J. M. Borwein and E. Wong, A survey of results relating to Giuga's
conjecture on primality, in {\it Advances in Mathematical Sciences:
{CRM}'s 25 years ({M}ontreal, {PQ}, 1994)}, CRM Proc. Lecture Notes,
Vol.\ 11, Amer. Math. Soc., 1997, pp. 13--27.

\bibitem{GIU}
G. Giuga, Su una presumibile propriet\'a caratteristica dei numeri primi, {\em Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. (3)} {\bf 14} (1950), 511--528.

\bibitem{VIC}
V. Ufnarovski and B. {\AA}hlander, How to differentiate a number, {\em J. Integer Seq.} {\bf 6} (2003), Article 03.3.4.
  
\bibitem{WIK}
Wikipedia, Giuga number, \url{http://en.wikipedia.org/wiki/Giuga_number}.

\end{thebibliography}

\bigskip
\hrule
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B99; Secondary 11A99.

\noindent \emph{Keywords: } 
arithmetic derivative, Giuga number, Lava's conjecture.

\bigskip
\hrule
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\noindent (Concerned with sequences
\seqnum{A003415} and
\seqnum{A007850}.)

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received  March 11 2011;
revised version received March 26 2012.  
Published in {\it Journal of Integer Sequences}, March 26 2012.

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