<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}

\usepackage{ifthen}
\usepackage{pgfpages}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

\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 
The Arithmetic Derivative and Antiderivative
}
\vskip 1cm
\large
Jurij Kovi\v{c}\\
Institute of Mathematics, Physics, and Mechanics \\
University of Ljubljana, Slovenia\\
\href{mailto:amadea.kovic@siol.net}{\tt amadea.kovic@siol.net} \\
\end{center}

\vskip .2 in
\begin{abstract}
The notion of the arithmetic derivative, a
function sending each prime to 1 and satisfying the Leibnitz rule,
is extended to the case of complex numbers with rational real and
imaginary parts. Some constraints on the solutions to some
arithmetic differential equations are found. The homogeneous
arithmetic differential equation of the $k$-th order is studied.
The factorization structure of the antiderivatives of natural
numbers is presented. Arithmetic partial derivatives are defined
and some arithmetic partial differential equations are solved.
\end{abstract}




\section{Introduction: Basic concepts}

The goal of this paper is: (i) to review what is known about the
arithmetic derivative \cite{Barb, Stay, Ufna}, (ii) to define some
related new concepts, and (iii) to prove some new results, mostly
related to the conjectures formulated in \cite{Barb, Ufna}.


\subsection{Definition of the arithmetic derivative}

Barbeau \cite{Barb} defined the arithmetic derivative as the
function $D: \mathbb Z \rightarrow \mathbb Z$, defined by the
rules:

$D(1) = D(0) = 0$

$D(p) = 1$ for any prime $p \in \mathbb P := \{2,3,5,7,\ldots,
p_{i},\ldots \}$.

$D(ab) = D(a)b + aD(b)$ for any $a,b \in \mathbb N$ (the Leibnitz
rule)

$D(-n) = -D(n)$ for any $n \in \mathbb N$.

Ufnarovski and \AA hlander \cite{Ufna} extended $D$ to rational
numbers by the rule:
$$       D(\frac{a}{b}) = (\frac{a}{b})' = \frac{a'b - ab'}{b^{2}}
$$

They also defined $D$ for real numbers of the form $x =
\prod_{i=1}^{k}p_{i}^{\, x_{i}} $, where $p_{i}$ are different
primes and $ x_{i}\in \mathbb Q$, by the additional rule:
$$D(x) = x' = x\sum_{i=1}^{k}\frac{x_{i}}{p_{i}}.$$

They showed that the definition of $D$ can be extended to all real
numbers and to the arbitrary unique factorization domain.

I extend the definition of the arithmetic derivative to the
numbers from the unit circle in the complex plane $\mathbb E =
\{e^{i\varphi} ,\varphi \in [0,2\pi] \} $ and to the Gaussian
integers ${\mathbb Z}[i] = \{a + bi, a,b \in \mathbb Z\}$.

It is not possible to extend the definition of the arithmetic
derivative to the Gaussian integers by the demand $D(q) = q' = 0$
for all irreducible Gaussian integers $q$ (the analog of primes).
If one wants the Leibnitz rule to still remain valid, one easily
gets a contradiction: $2 = (1 + i)(1 - i)$, hence $D(2) = 1\cdot(1
- i) + (1 + i)\cdot 1 = 2$, but $D(2) = 1$ because $2$ is a prime.

However, one can extend $D$ to ${\mathbb Q}[i] = \{a + bi, a,b \in
\mathbb Q\}$ using the polar decomposition of complex numbers as
follows:

\begin{proposition} \label{prop1} i) For any $\varphi = (m/n)\pi$,
where $m,n \in \mathbb Q$ and $n \neq 0$, the equation
$D(e^{i\varphi }) = 0$ holds.

ii) Let $\mathbb E = \{z \in \mathbb C; |z| = 1 \} $ be the unit
circle in the complex plane. There are uncountably many functions
$D: \mathbb E \rightarrow \mathbb E$ satisfying the condition
$D(zw) = D(z)w + zD(w)$ for all $z,w \in \mathbb E$.

iii) If one defines the arithmetic derivative on $\mathbb E$ as
follows: $D(e^{i\varphi}) = 0 $ for all $e^{i\varphi} \in \mathbb
E$, then the definition of the arithmetic derivative can be
uniquely extended to ${\mathbb Q}[i]$ so that the Leibnitz rule
and the quotient rule remain valid.
\end{proposition}
\begin{proof} i) Take any $\varphi = (m/n)\pi$, where $m,n \in \mathbb Z$
and $n \neq 0$. Then $e^{in\varphi} = \pm1$, hence
$D(e^{in\varphi}) = 0$. The Leibnitz rule implies
$D(e^{in\varphi}) = D(e^{i\varphi})(e^{i\varphi} )^{n-1}$ and
because $(e^{i\varphi} )^{n-1}\neq 0 $ it must be that
$D(e^{i\varphi}) = 0$.

ii) Sketch of the proof: One can easily see that there is a 1-1
correspondence between the functions $D: \mathbb E \rightarrow
\mathbb C$ satisfying the Leibnitz rule $D(zw) = D(z)W + zD(w)$
and the additive functions $L: \mathbb R \rightarrow \mathbb C$
with a period of $2\pi$ (the correspondence is given by the
formulas $L(s) := \frac{D(e^{is})}{e^{is}}$ and $D(e^{is}) :=
L(s)e^{is}$).

It is also easy to see that for any additive function (satisfying
Cauchy's functional equation $L(s + t) = L(s) + L(t)$) it is:
$L(qx) = q \cdot L(x) $ for all $q \in \mathbb Q$ and $x \in
\mathbb R$. Because $\mathbb R$ and $\mathbb C$ are linear spaces
over $\mathbb Q$, this means that every additive function $L:
\mathbb R \rightarrow  \mathbb R$ is a $\mathbb Q$-linear
transformation. Additive functions from $\mathbb R$ into $\mathbb
C$ with a period of $2\pi$ are therefore in a 1-1 correspondence
with $\mathbb Q$-linear transformations $L: \mathbb R \rightarrow
\mathbb C$ such that $L(2\pi) = 0$.

A linear transformation is uniquely determined with its values on
any of the bases of its domain. Let $B$ be a base of the space
$\mathbb R$ such that $\pi \in B$ (its existence is guaranteed by
Zorn's lemma). Thus the $\mathbb Q$-linear transformations $L:
\mathbb R \rightarrow \mathbb C$ such that $L(2\pi) = L(\pi) = 0$
are in a 1-1 correspondence with the set of all functions $L: B
\rightarrow \mathbb C$ such that $L(\pi) = 0$.

The base $B$ is obviously not countable, hence the same also holds
for the set of all functions $L: B\setminus \{\pi\} \rightarrow
\mathbb C$. By the chain of proven 1-1 correspondences, this also
holds for the set of all functions $D: \mathbb E \rightarrow
\mathbb E$ satisfying the Leibnitz rule.


iii) If one writes such numbers in their polar form $a + bi =
re^{i\varphi}$, then $D(re^{i\varphi}) = D(r)e^{i\varphi} +
rD(e^{i\varphi} )$ $=D(r)e^{i\varphi} = D((a^{2} +
b^{2})^{1/2})e^{i\varphi}$ because $D(e^{i\varphi}) = 0$ and $r =
(a^{2} + b^{2})^{1/2}$. The Leibnitz rule is still valid because
for any $z = re^{i\varphi }$ and $w = se^{i\psi } $ it is: $D(zw)
= D((re^{i\varphi })(se^{i\psi} )) = D(rs)e^{i\varphi + \psi} =
(D(r)s + r(D(s))e^{i\varphi + \psi} = $ $D(r)e^{i\varphi}
se^{i\psi} + re^{i\varphi}D(s)e^{i\psi } = D(z)w + zD(w). $ The
quotient rule is also still valid because $\frac{D(z)w -
zD(w)}{w^{2}} = \frac{D(r)e^{i\varphi}se^{i\psi } -
re^{i\varphi}D(s)e^{i\psi }}{s^{2}e^{i2\varphi}} = \frac{(D(r)s -
rD(s))e^{i\varphi}}{s^{2}e^{i\psi }}  = D(\frac{r}{s})e^{i(\varphi
- \psi)} = D(\frac{r}{s}e^{i(\varphi - \psi)})=
D(\frac{re^{i\varphi}}{se^{i\psi}}) = D(\frac{z}{w})$.
\end{proof}



Stay \cite{Stay} generalized the concept of the arithmetic
derivative still further, practically for any number, using
advanced techniques such as exponential quantum calculus. In this
paper I focus on the arithmetic derivative as a function defined
for natural numbers and rational numbers.

\subsection{Higher derivatives and the logarithmic derivative}

\emph{Higher derivatives} $n^{(k)}$ are defined inductively:
$n^{(2)} = D^{2}(n) = D(D(n)) = n'' = (n')'$, $n^{(k+1)} =
D^{k+1}(n) = D(D^{k}(n))$. Many conjectures on the arithmetic
derivative focus on the behavior of sequences $(n, n', n'',\ldots
) $.

Ufnarovski and \AA hlander \cite{Ufna} conjectured that for each
$n \in \mathbb N$ exactly one of the following can happen: either
$n^{(k)} = 0$ for sufficiently large $k$, or $lim _{k\rightarrow
\infty} = \infty$, or $n = p^{p}$ for some prime $p$ (in this case
$n^{(k)} = n$ for each $k \in \mathbb N$). They introduced the
function $L$, called the \emph{logarithmic derivative}, satisfying
the condition $$L(x) = \frac{x'}{x} = \frac{D(x)}{x}.$$ For any $x
= \prod_{i=1}^{k}p_{i}^{\, x_{i}} $, where $p_{i}$ are different
primes and $ x_{i}\in \mathbb Q$, $L$ satisfies the condition
$$L(x) = \sum_{i=1}^{k}\frac{x_{i}}{p_{i}}.$$

For every prime $p$ and every $m,n \in \mathbb N$ the following
formulas hold: $L(p) = \frac{1}{p},$ $L(p^{\frac{m}{n}}) =
\frac{m}{np}, $ $L(1) = 0.$ From the definition of the logarithmic
derivative also follow the formulas $L(-x) = L(x), L(0) = \infty$
and $D(x) = L(x)\cdot x $ \cite{Ufna}, hence $D^{2}(x) = D(L(x))x
+ L(x)D(x)$. I also use the notation $L(x) = x^{*} $.

The logarithmic derivative is an additive function: $L(xy) = L(x)
+ L(y)$ for any $x, y \in \mathbb
 Q$. Consequently, using a table of values $L(p) = \frac{1}{p}$ (computed to
 sufficient decimal places!) and the formula $D(x)
= L(x)\cdot x$, it is easy to find $D(n)$ for $n \in \mathbb
 N $ having all its prime factors in the table. For example, $D(5\cdot11^{3}
 ) = L(5\cdot11^{3})\cdot5\cdot11^{3}$ and
 because $ L(5\cdot11^{3}) = L(5) + 3L(11) = 0.2000 + 3\cdot0.0909 = 0.4727$, one can
 calculate $D(5\cdot11^{3}) = \lceil0.4727\cdot6655 \rceil
 = \lceil3145.8185 \rceil = 3146.$



{%\small
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  $P_{N}$ &amp;  $2$&amp; $3$&amp; $5$&amp; $7$&amp; $11$ &amp;$13$&amp; $17$  \\
  \hline
  $\frac{1}{P_{N}}$ &amp; $0.5000$&amp; $0.3333$&amp; $0.2000$&amp; $0.1429$&amp; $0.090$&amp; $0.0769$ &amp;$0.0588$  \\

   \hline
\end{tabular}
\end{center}
}

\begin{center}
Table 1: Values of the logarithmic derivative  for the first seven
primes
\end{center}

\section{Brief review of known results and conjectures}\label{chap2}

Barbeau \cite{Barb} proved that if $n$ is not a prime or unity
then $D(n) \geq 2\sqrt{n}$ with equality only if $n = p^{2}, p \in
\mathbb P$. He showed that, for integers possessing a proper
divisor of the form $p^{p}, p \in \mathbb P$, $\lim _{k\rightarrow
\infty} D^{k}(n) = \infty.$

Ufnarovski and \AA hlander \cite{Ufna} translated some famous
conjectures in number theory (e.g., the Goldbach conjecture, the
Prime twins conjecture) into conjectures about the arithmetic
derivative. They formulated many other conjectures, mostly related
to (arithmetic) differential equations; for example that the
equation $x' = 1$ has only primes as positive rational solutions
(but it has a negative rational solution $x = -\frac {5}{4}$).

They also conjectured that the equation $n'' = n$ has no other
solutions than $n = p^{p}, p \in \mathbb P$ in natural numbers and
that there are some rational numbers without antiderivatives (or
"integrals"). I study these conjectures in Section~\ref{chap3}.



\subsection{Arithmetic differential equations and integrals}

Most of the known results by Ufnarovski and \AA hlander
\cite{Ufna} focus on arithmetic differential equations of the
first and second order. For example:

The only solutions to the equation $n' = n$ in natural numbers are
$n = p^{p}$, where $p$ is any prime.

The nonzero solutions to $x' = 0$ are the rational numbers of the
form: $x = \pm\prod p_{i}^{\alpha_{i}p_{i}} $, where $p_{1},
\ldots ,p_{k} $ are different primes and $\{\alpha_{1},\ldots,
\alpha_{n}\}$ is a set of integers such that $\Sigma_{i=1}^{k}
\alpha_{i} = 0$.

The rational solutions to differential equations $x' = x\alpha$
for all rational numbers $\alpha = a/b$, where $\gcd(a,b) = 1$ and
$b &gt; 0$, are all of the form $x = x_{0}y$, where $x_{0}$ is a
nonzero particular solution and $y$ is any rational solution of
the equation $y' = 0 $.

Let $I(a)$ denote all the solutions of differential equation $n' =
a$ for $n \in \mathbb N$ and let $i(a)$ denote the number of such
solutions, called the integrals of $n$. Because $n'\geq 2\surd n $
if $n$ is not a prime or unity \cite{Barb}, the solutions satisfy
$n \leq \frac {a^{2}}{4} $, hence $i(a)&lt; \infty$ for any $a \in
\mathbb N$. Because $\gcd(n,n') = 1$ if and only if $n$ is
square-free, all integrals of primes are products of different
primes: $p_{1}\cdots p_{k} $. It will be seen
(Corollary~\ref{coro1}) that there are primes without integrals
(e.g., primes 2, 3, 17). If $D(n)$ were known for each $n \in
\mathbb N$, one would know which natural numbers are primes
(because the equation $n' = 1$ has only primes as solutions in
natural numbers). Ufnarovski and \AA hlander \cite{Ufna} gave a
list of all $a \leq 1000 $ having no integral, a list of those
numbers $a \leq 100$ having more than one integral, and a list of
those $a \leq 100$ for which $i(a) = 1$. It will be seen
(Corollary~\ref{coro2}) that something can also be said about the
possible factorization structure of the integrals of a given
natural number.

\section{New results}\label{chap3}

Ufnarovski and \AA hlander \cite{Ufna} solved the equation $x' =
\alpha x$ in rational numbers for every rational number $\alpha$.
Nonetheless it is interesting to know which natural numbers solve
this equation when $\alpha = m$ is a natural number.


\begin{proposition} \label{prop2}
Let $m \in \mathbb N$. The solutions to the equation $x' = mx$ in
natural numbers $x$ are exactly the numbers of the form $x =
p_{1}^{p_{1}n_{1}} \cdots p_{k}^{p_{k}n_{k}}$, where $n_{1} +
\dots + n_{k} = m.$
\end{proposition}
\begin{proof} Let $x = p_{1}^{e_{1}} \cdots p_{k}^{e_{k} }$ be the
factorization of $x$. The condition $x' = L(x)x = mx$ implies
$L(x)(p_{1}\cdots p_{k} ) = (\frac{e_{1}}{p_{1}}+ \cdots +
\frac{e_{k}}{p_{k}})(p_{1}\cdots p_{k} ) = m(p_{1}\cdots p_{k} ).$
Because the right side of this equation is divisible by any of the
primes $p_{1}, \ldots ,p_{k}$, the left side must also be. This is
possible only if for every prime $p_{i}$ the corresponding
exponent $e_{i}$ is the multiple of this prime $e_{i} =
n_{i}p_{i}$. The derivative of such a number $x =
p_{1}^{p_{1}n_{1}} \cdots p_{k}^{p_{k}n_{k}} $ is $D(x) =
D(p_{1}^{p_{1}n_{1}} \cdots p_{k}^{p_{k}n_{k}} ) = (n_{1} + \cdots
+ n_{k})x$ (by the Leibnitz rule and because $D(p^{p_{i}n_{i}}) =
n_{i}p^{p_{i}n_{i}})$ and this is equal to $mx$ if and only if
$n_{1} + \cdots + n_{k} = m.$
\end{proof}


\subsection{The homogeneous differential equation of the $k'$th order}

What are the solutions $x$ of the differential equation
$$a_{k}x^{(k)} + a_{k-1}x^{(k-1)} + \cdots + a_{2}x^{(2)} + a_{1}x' + a_{0}x = 0$$
with rational coefficients $a_{i}$? In order to answer this
question I introduce the concept of a logarithmic class.

\begin{definition}\emph{Let $A$ be any chosen subring of the ring of complex
numbers for which the arithmetic derivative is defined. The
logarithmic class $N_{r,A}$ of the number $r$ consists of all
numbers $x \in A$ with the same logarithmic derivative: $N_{r,A} =
\{x \in A; x^{*} = L(x) = r\} $}.
\end{definition}

\begin{remark} If it is made perfectly clear which $A$ is being worked with the
shorter notation $N_{r}$ can be used. For the purpose of this
article, let $A$ be the set of all complex numbers with rational
real and imaginary parts: $A = {\mathbb Q}[i] = \{a + bi, a,b \in
\mathbb Q\}$.
\end{remark}

\begin{example} Besides the number $0$ and the complex numbers $z = e^{i\varphi}
\in A$ on the unit circle in the Gaussian plane the class $N_{0}$
also contains rational solutions $x = \pm\prod
p_{i}^{\alpha_{i}p_{i}} $, where $p_{1}, \ldots ,p_{k} $ are
different primes and $\{\alpha_{1},\ldots, \alpha_{n}\}$ is a set
of integers such that $\Sigma_{i=1}^{k} \alpha_{i} = 0$; for
instance, $x = \frac{4}{27}$ \cite{Ufna}. All $x \in N_{0}$ solve
the equation $x' = 0$. \end{example}

The class $N_{1,\mathbb Q}$ contains all numbers $p^{p}$, where
$p$ is any prime because $D(p^{p}) = p^{p}$. The class
$N_{-1,\mathbb Q}$ contains all numbers $p^{-p} $, where $p$ is
any prime because $D(p^{-p} ) = -p^{-p}. $ Do $N_{1,\mathbb Q}$
and $N_{-1,\mathbb Q}$ also contain other rational solutions? Yes:

\begin{proposition}\label{prop3} If $(\frac{a}{b})' = \pm\frac{a}{b} \in \mathbb Q$
and if $a$ and $b$ have no common factors greater than 1, then $a
= p_{1}^{p_{1}n_{1}} \cdots p_{k}^{p_{k}n_{k}}$, $b =
q_{1}^{q_{1}s_{1}} \cdots q_{k}^{q_{k}n_{s}}$, and $a^{* }- b^{*}
= \pm 1$, where $n_{i}, m_{j}, k,l $ are nonnegative integers and
$a^{*}, b^{*}$ logarithmic derivatives. All such numbers are
solutions because: $(\frac{a}{b})'= (\frac{a}{b})^{*}(\frac{a}{b})
= \pm(\frac{a}{b}).$
\end{proposition}
\begin{proof} If $(\frac{a}{b})' = \pm\frac{a'b - b'a}{b^{2}} =
\pm\frac{a}{b} $ then $a'b - b'a = \pm ab$, hence $\gcd(a,b) = 1$
implies $a' = ma$ and $b' = nb$, where $m = a^{*}$ and $n = b^{*}$
are natural numbers. Hence (by Proposition~\ref{prop2}) $a =
p_{1}^{p_{1}n_{1}} \cdots p_{k}^{p_{k}n_{k}}$, where $n_{1} +
\cdots + n_{k} = m,$ and $b = q_{1}^{q_{1}s_{1}} \cdots
q_{k}^{q_{k}n_{s}}$, where $s_{1} + \cdots + s_{k} = n$. Because
$(\frac{a}{b})' = \pm\frac{a}{b} $ implies $ (\frac{a}{b})^{*} =
\pm 1$, it must be $a^{*} - b^{*} = \pm 1.$
\end{proof}

\begin{proposition} \label{prop4} i) The derivative $D$ sends logarithmic classes into logarithmic
classes as follows: $D(N_{r}) \subseteq N_{r^{*} + r}$.
Consequently $D(N_{r}) \subseteq N_{r}$ if and only if $r^{*} = 0$
and therefore $r' = 0$.

ii) If $r^{*} + r = 0$ and $r \neq 0$ then $D(N_{r}) = N_{0} -
\{0\} \neq N_{0}$. So in this case $D(N_{r}) \supseteq N_{r^{*} +
r}$ is not true.

iii) If $r^{*} + r \neq 0$ then $D(N_{r}) = N_{r^{*} + r}$.

\end{proposition}
\begin{proof}

If $r = 0$ then $N_{r} = N_{0} = N_{r^{*} + r}$. So let us now
assume that $r\neq  0$.

i) $D(N_{r}) \subseteq N_{r^{*} + r}$ is true because $x \in
N_{r}$ implies $x'= rx$, hence $(x')' = r'x + rx' = \frac{r'x'}{r}
+ rx' = (\frac{r'}{r} + r)x' = (r^{*} + r )x' $.

ii)  It is possible indeed that $r^{*} + r = 0$ while $r \neq 0$
(an example is $r = \frac {1}{5}, r' = -\frac {1}{5^{2}}, r^{*} =
- \frac {1}{5})$. If $r^{*} + r = 0$ then $0 \in N_{r^{*} + r}$
because $0' = 0$. Now suppose there is an $x \in N_{r}$ such that
$D(x) = 0$. Wowever, this implies $x \in N_{0} $, hence $r = 0$
and there is a contradiction with the assumption $r \neq 0$. This
means that in this case it is not true $D(N_{r}) \supseteq
N_{r^{*} + r} = N_{0}$ because there is no $x \in N_{r}$ such that
$D(x) = 0$.

Now let it be proved that for each nonzero $y \in N_{r^{*} + r}$
there is an $x \in N_{r}$ such that $D(x) = y$. If there is any
such $x \in N_{r}$, then it must be $y = x' = rx$, hence there is
at most one such $x$ and it is defined by the formula $x =
\frac{y}{r}$. Now suppose the derivative of this $x$ is not equal
to $y$. This assumption leads to a contradiction because
$D(\frac{y}{r}) = \frac{D(y)r - yD(r)}{r^{2}} = \frac{(r^{*} +
r)yr - yr^{*}r}{r^{2}} = \frac{yr^{2}}{r^{2}}\neq y$ implies $1
\neq 1$. Because $D(x) = y = rx$, this $x$ is indeed a member of
$N_{r}$. Thus it has been proved that $D(N_{r}) \supseteq N_{0} -
\{0\}$.

Because it is already known that i) is true and because it has
been shown that there is no $x \in N_{r}$ such that $D(x) = 0$,
the equation $D(N_{r}) = N_{0} - \{0\}$ holds. Moreover, because
$0 \in N_{0}$, it is also true that $N_{0}$ is a proper subset of
$N_{0} - \{0\}$.

iii) If $r^{*} + r \neq 0$ then $r \neq 0$, and $N_{r^{*} + r}$
contains only nonzero elements because $0 \in N_{0} $. Now it is
possible to repeat the reasoning as in ii) and for any $y \in
N_{r^{*} + r}$ one finds an $x \in N_{r}$ such that $y = D(x)$.
Hence $D(N_{r}) \supseteq N_{r^{*} + r}$ and this together with i)
implies $D(N_{r}) = N_{r^{*} + r}$.\end{proof}

\begin{remark} It is already known that there are many rational solutions to the
equation $r' = 0$; for example, $r = 1, -1, 0,$ hence: $D(N_{1})
\subseteq N_{1}, D(N_{-1}) \subseteq N_{-1}$, $D(N_{0}) \subseteq
N_{0}.$ \end{remark}

\begin{proposition} \label{prop5}
All the derivatives $x^{(k)}$ of any number $x$ can be expressed
as functions of the logarithmic derivative $L(x) = x^{*}$ as
follows: $x^{(k)} = f_{k}(x^{*})x$ where $f_{1}(x^{*}) = x^{*} $
and $f_{k+1}(x^{*}) = ((f_{k}(x^{*}))' + f_{k}(x^{*}))x^{*}.$ Thus
$x' = x^{*}x, x^{(2)} = ((x^{*})' + (x^{*})^{2})x$ etc.
\end{proposition}
\begin{proof} This is true for $k = 1$. Suppose $x^{(k)} =
f_{k}(x^{*})x$. Then by the Leibnitz rule: $x^{(k+1)} =
(f_{k}(x^{*}))'x + f_{k}(x^{*})x' = ((f_{k}(x^{*}))' +
f_{k}(x^{*})x^{*})x.$ \end{proof}

\begin{proposition} \label{prop6}
Any homogeneous differential equation $f(x) \equiv a_{k}x^{(k)} +
a_{k-1}x^{(k-1)} + \cdots + a_{2}x^{(2)} + a_{1}x' + a_{0}x = 0$
reduces to an equation $g(x^{*}) = 0$. Consequently, if the set of
nontrivial solutions to any homogeneous differential equation
$f(x) = 0$ is not empty, then it consists of some classes $N_{r}$.
\end{proposition}
\begin{proof} Let $r = x^{*} = \frac{x'}{x}.$ Because $x^{(k)} =
f_{k}(x^{*})x$, one can divide the differential equation by $x$
and get an equation of the form $g(r) = 0$. Thus $f(x) = 0$ if and
only if $g(x^{*}) = 0.$ Hence whether $f(x) = 0$ or not depends
only on the logarithmic class $N_{r} = N_{x^{*}}$. Note also that
the degree of the polynomial $g$ is $k - 1$, one less than the
degree of the polynomial $f$.
\end{proof}

\begin{example} The nonzero solutions to the equation $x'' - x = 0$ implying
$((x^{*})' + (x^{*})^{2})x  - x = 0$ satisfy the equation
$((x^{*})' + (x^{*})^{2} - 1 = 0$ or $(x^{*})' = 1 - (x^{*})^{2}
$. Thus the nonzero solutions to $x'' - x = 0$ exist if and only
if the nonhomogeneous equation $r' = 1 - r^{2}$ can be solved.
\end{example}

\begin{proposition} \label{prop7} For every $r \in \mathbb Q$ the class $N_{r}$ is not empty because it contains the numbers $p^{pr}$, where $p$ is any prime.
\end{proposition}
\begin{proof}  $L(p^{pr}) = r$ for any prime $p$, hence $p^{pr}
\in N_{r}$. \end{proof}

Thus in solving the homogenous differential equation $f(x) = 0$
one can always search for the solutions of the form $x = p^{pr}$.
Of course, other solutions are also possible.

\begin{remark} Numbers $p^{pr}$ behave in the first
derivative just like the exponential function $e^{xr}$ whose
derivative is $re^{xr}$. However, for the higher derivatives the
analogy is no longer valid because $(p^{p}r)'' = (r' +
r^{2})p^{pr}$, while $(e^{xr})'' = r^{2}e^{xr}$. This is one of
the reasons why solving an arithmetic differential equation is
harder than solving the analogous problem for functions.
\end{remark}

\begin{proposition}\label{prop8} i) In addition to the trivial solutions $x \in N_{0}$
the equation $x'' = 0$ also has solutions $x \in N_{1/p}$, where
$p$ is any prime.

ii) All solutions $x \in N_{r}$ of $x'' = 0$ satisfy the
condition: $r^{*} + r = 0$ (hence $r \in N_{-r}$ and $r' = r^{*}r
= -r^{2} $) and all such $x$ are solutions to $x'' = 0$. If $r =
\frac{a}{b} \in \mathbb Q$ then $\frac{a}{b} = b^{*} - a^{*} =
(\frac{b}{a})^{*}$, hence $(\frac{b}{a})' = 1$.
\end{proposition}
\begin{proof} i) If $x' = \frac{1}{p}$ then $x'' = - \frac{1}{p^{2}}x
+ \frac{1}{p}\frac{1}{p}x = 0 = (x')^{*}x'$, hence $(x')^{*} = 0
$.

ii)  Any solutions $x \in N_{r}$ of $x'' = 0$ satisfy the
condition $D^{2}(N_{r}) \subseteq D(N_{r^{*} + r}) = D_{0}$, hence
$r^{*} + r = 0$. Conversely, $r^{*} + r = 0 $ implies
$D^{2}(N_{r}) \subseteq D(N_{r^{*} + r}) = D({N_{0}}) = \{0\}$,
hence $x'' = 0$. If $r = \frac{a}{b}$ then $r' = \frac{a'b -
ab'}{b^{2}} = \frac{a'b - ab'}{ab}\frac{a}{b}$, hence
$(\frac{a}{b})^{*} = \frac{a'b - ab'}{ab} = -r = -\frac{a}{b}$,
therefore $\frac{a'}{a} - \frac{b'}{b} = -\frac{a}{b}$ and
$\frac{a}{b} = b^{*} - a^{*} = (\frac{b}{a})^{*} $. Hence
$(\frac{b}{a})' = \frac{b}{a}^{*}\frac{b}{a} =
\frac{a}{b}\frac{b}{a} = 1$.
\end{proof}

\begin{example} Ufnarovski and \AA hlander \cite{Ufna}
observed that if $x = -\frac{5}{4}$ then $x' = 1$ (hence $x'' = 0
$). Then $r = x^{*} = - \frac{4}{5}$, $r' = \frac{-4\cdot5 -
(-4)\cdot1}{5^{2}} = -\frac{16}{25} = \frac{4}{5}\frac{-4}{5},$
hence $r^{*} = \frac{4}{5}$ and $r^{*} + r =  \frac{4}{5} +
(-\frac{4}{5} ) = 0$. Hence all $x \in N_{-\frac{4}{5}}$ solve
$x'' = 0$. \end{example}

\subsection{The graph of derivatives of natural numbers}

Of interest is the structure of the infinite directed graph
$G_{D}$, whose vertices correspond to natural numbers $n$ and
whose arcs $n\rightarrow D(n)$ connect the number and its
derivative. The corresponding dynamic system: $n \rightarrow D(n)$
has two obvious attractors: $0$ and $\infty$. There are numbers
$n$ with an increasing sequence of derivatives $n&lt; n'&lt; n''&lt; \cdots
&lt; n^{(k)} &lt; n^{(k+1)} &lt; \cdots $ (e.g., $n = p^{pk}, p \in \mathbb
P)$, so there are paths of infinite length in the graph $G_{D}$.

Ufnarovski and \AA hlander \cite{Ufna} conjectured that the
equation $n^{(k)} = n$ has only trivial solutions $p^{p}$, where
$p \in \mathbb P$, satisfying $n' = n $. If this is true then the
only cycles in $G_{D}$ are the loops in these fixed points. They
have shown that if $m' = n$ and $n' = m$ then $m$ and $n$ must be
square-free numbers: $n = \prod_{i=1}^{k} p_{i} $ and $m =
\prod_{j=1}^{l}q_{j} $, where all $p_{i}$ are distinct from all
$q_{i}$. I present further constraints
(Propositions~\ref{prop9},~\ref{prop10},~\ref{prop11},~\ref{prop11a})
on the structure of possible solutions of the equation $n'' = n$
in natural numbers.

However, the equation $x'' = x$ has non-trivial rational solutions
of the form $x = p^{-p}$ where $p \in \mathbb P$ and they also
satisfy the equation $x' = -x.$

Let us first show that any eventual nontrivial solutions $m,n$
(different from $m = n = p^{p}$ where $p \in \mathbb P$) of the
system $n' = m$, $m' = n$ in natural numbers cannot be just a
product of two primes; at least one of the numbers $m,n \in
\mathbb N$ solving such a system must have at least three
different prime factors.

\begin{proposition} \label{prop9} The system $n' = m$, $m' = n$ has no solutions
in natural numbers of the form $m = p_{1}p_{2}$ and $n =
q_{1}q_{2} $, where $p_{1},p_{2},q_{1},q_{2} \in \mathbb P $.
\end{proposition}
\begin{proof} Because $p_{1},p_{2},q_{1},q_{2}$ must be
different primes, it is not possible for both $m$ and $n$ to have
the factor 2. Therefore one can assume that $p_{1}$ and $p_{2}$
are both odd. Then $m' = p_{1} + p_{2} = n = 2q_{2}$ and $n' = 2 +
q_{2} = m = p_{1}p_{2}$, hence $2q_{2} =  2p_{1}p_{2} - 4 = p_{1}
+ p_{2}$ and $p_{1} = \frac{p_{2} + 4}{2p_{2} - 1} = 1 + \frac{5 -
p_{2}}{2p_{2} - 1}$, and this implies $p_{2} \leq 5$ because
$p_{1}$ must be a natural number. Moreover, it was assumed that
$p_{2}$ is odd. However, for $p_{2} = 3$ one would get $p_{1} = 1
+ \frac{2}{5} = \frac{7}{5} $ and for $p_{2} = 5 $ one would get
$p_{1} = 1$. \end{proof}

A computer search showed that there are no natural solutions to $
x'' = x$ such that $x &lt; 10000 $. This result can be improved, at
least if the smaller of the numbers $m$ and $n$ is odd.

\begin{proposition} \label{prop10} Let $n' = m$, $m' = n$ and let $n = 2j - 1 &lt; m.$ Then there are at
least nine primes in the factorization of $n$ and $n &gt; 3\cdot 5
\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29$.
\end{proposition}
\begin{proof} It is known that $m$ and $n$ must be products of
different primes  $n = \prod_{i=1}^{k} p_{i} $ and $m =
\prod_{j=1}^{l}q_{j} $, where all $p_{i}$ are distinct from all
$q_{i}$. Because $n &lt; m = n' = L(n)n$ we have $L(n) =
\sum_{i=1}^{k}\frac{1}{p_{i}}
&gt; 1$. Because $n$ is odd, all $p_{i}$  are greater than 2. The sum of the reciprocals
of the first eight odd primes is: $\frac{1}{3} + \frac{1}{5} +
\frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \frac{1}{17} +
\frac{1}{19} + \frac{1}{23} = 0.9987\cdots &lt; 1$. Thus $k \geq 9 $
and $n
&gt; 3\cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23
\cdot 29$. \end{proof}

If $n = 2j &lt;m$ then the sum $\frac{1}{2} +\frac{1}{3} +\frac{1}{5}
$ is already greater than 1 and one does not get any such
estimate.

Another possible approach to the system $n' = m, m' = n$ in
natural numbers is based on the comparison of a square-free number
and its derivative; this comparison will imply some inequalities
for the smallest primes and biggest primes in the factorizations
of $n$ and $m$.

\begin{proposition} \label{prop11} Let $n = p_{1}\cdots p_{r}$, where $p_{1}&lt; \cdots
&lt;p_{r}$, and let $ m = q_{1}\cdots q_{s}$ where $q_{1}&lt; \cdots
&lt;q_{s}$ be square-free numbers such that $n' = m, m' = n$. Then: $
p_{1}q_{s} &lt; rs \leq (\frac{r + s}{2})^{2}$, $q_{1}p_{r} &lt; rs \leq
(\frac{r + s}{2})^{2}$, $p_{1} &lt; r,$ $q_{1} &lt; s.$ As a consequence
$n$ and $m$ must together have at least 34 prime factors $p_{i}$
and $q_{j}$, thus: $r + s \geq 34$. Hence at least one of $m$ and
$n$ has at least 17 prime factors and is not smaller than the
product of the first 17 primes: $2 \cdot 3 \cdot 5 \cdot 7 \cdot
11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 27 \cdot 31 \cdot 37
\cdot41 \cdot 43 \cdot 47 \cdot 53 \cdot 59$. If
$\min\{p_{1},q_{1}\} \geq 3$, then $r + s \geq 57$, hence at least
one of $m$ and $n$ has at least 29 prime factors. If
$\min\{p_{1},q_{1}\} \geq 5$, then $r + s \geq 110$, hence at
least one of $m$ and $n$ has at least 55 prime factors.

\end{proposition}
\begin{proof}  Let $n = p_{1}\cdots p_{r}$ and $ m = q_{1}\cdots
q_{s}$ be square-free numbers such that $n' = m, m' = n$. Then $0
&lt; r\frac {n}{p_{r}} &lt; n' &lt; r\frac {n}{p_{1}}$ and $0 &lt; s\frac
{m}{q_{s}} &lt; m' &lt; s\frac {m}{q_{1}}.$ Hence: $ p_{1}q_{s} &lt; rs $
and $q_{1}p_{r} &lt; rs.$ Because $r &gt; 1, s
&gt; 1 $ we have $p_{r} &gt; r$ and $q_{s} &gt; s$, hence $p_{1}&lt; r$ and
$q_{1} &lt; s.$ Let $r + s = N$. Then $rs \leq (\frac{r + s}{2})^{2}
= (\frac{N}{2})^{2}$. Thus $2q_{s} \leq p_{1}q_{s} &lt;
(\frac{N}{2})^{2} $ and $2p_{r} \leq q_{1}p_{r}
&lt;(\frac{N}{2})^{2},$ hence $ 2\max\{p_{r},q_{s}\}
&lt;\frac{N^{2}}{4}$.

Let $P_{i}$ denote the $i$-th prime (thus $P_{1} = 2$, $P_{2} =
3$, $P_{3} = 5$, etc.) Because all the primes $p_{1},
\ldots,p_{r},q_{1}, \ldots ,q_{s}$ are distinct, we have
$\max\{p_{r},q_{s}\} \geq P_{r+s } = P_{N}.$ Thus $2P_{N} =
2P_{r+s} \leq 2\max\{p_{r},q_{s}\} &lt; rs \leq \frac{N^{2}}{4}.$

It is possible to directly check that $P_{N}  \geq
\frac{N^{2}}{8}$ if $ N\leq 33$.

{%\small
\begin{center}\label{table1}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  N &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$&amp; $6$&amp; $7$&amp; $8$&amp; $9$&amp; $10$&amp; $11$&amp; $12$\\
  \hline
  $P_{N}$ &amp; $2$ &amp; $3$ &amp; $5$ &amp; $7$ &amp; $11$&amp; $13$&amp; $17$&amp; $19$&amp; $23$&amp; $27$&amp; $31$&amp; $37$\\
$N^{2}$ &amp; $1$ &amp; $4$ &amp; $9$ &amp; $16$ &amp; $25$&amp; $36$&amp; $49$&amp; $64$&amp; $81$&amp; $100$&amp; $121$&amp; $144$ \\
   \hline
\end{tabular}
\end{center}
}

{%\small
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  N &amp;$13$ &amp; $14$ &amp; $15$ &amp; $16$ &amp; $17$ &amp; $18$ &amp; $19$ &amp; $20$&amp; $21$&amp; $22$&amp; $23$ \\
  \hline
  $P_{N}$ &amp; $41$ &amp; $43$ &amp; $47$ &amp; $53$ &amp; $59$&amp; $61$&amp; $67$&amp; $71$&amp; $73$&amp; $79$&amp; $83$\\
$N^{2}$ &amp;  $169$&amp; $196$&amp;  $225$&amp; $256$ &amp; $289$ &amp; $324$ &amp; $361$ &amp; $400$&amp; $441$&amp; $484$&amp; $529$  \\
   \hline
\end{tabular}
\end{center}
}

{%\small
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  N &amp;  $24$&amp; $25$&amp; $26$&amp; $27$&amp; $28$ &amp;$29$&amp; $30$ &amp; $31$ &amp; $32$ &amp; $33$ &amp; $34$\\
  \hline
  $P_{N}$ &amp; $89$&amp; $97$&amp; $101$&amp; $103$&amp; $107$&amp; $109$ &amp;$113$ &amp;$127$ &amp;$131$ &amp;$137$ &amp;$139$\\
$N^{2}$ &amp;  $576$&amp; $625$&amp; $676$&amp; $729$ &amp;$784$&amp; $841$&amp; $900$ &amp; $961$ &amp; $1024$ &amp; $1089$ &amp; $1156$\\
   \hline
\end{tabular}
\end{center}
}

\begin{center}
Table 2: The first 34 primes $p_{N}$ and squares $N^{2}$
\end{center}

Hence it must be that $N \geq 34.$ If $\min\{p_{1},q_{1}\} \geq
3$, one can get a much better estimate from the condition: $p_{N}
&lt; \frac{N^{2}}{12}$, which is first fulfilled when $N \geq57  $.
If $\min\{p_{1},q_{1}\} \geq 5$, then $p_{N} &lt; \frac{N^{2}}{20} $,
which is first fulfilled when $N \geq 110$. \end{proof}

\begin{proposition}\label{prop11a} Suppose $n' = m$ and
$n' = m$ and $n,m$ are both odd. Let $n$ have $r_{1}$ primes
$p_{i}\equiv 1$ (mod 4) and $s_{1} $ primes $q_{j}\equiv -1$ (mod
4) and let $m$ have $r_{2}$ primes $p_{i}\equiv 1$ (mod 4) and
$s_{2} $ primes $q_{j}\equiv -1$ (mod 4). Then
$(r_{1}-s_{1})(r_{2}-s_{2}) \equiv 1$ (mod 4).
\end{proposition}
\begin{proof}
Barbeau \cite[pp.\ 121--122]{Barb} proved that if $n = p_{1}\cdot
p_{2}\dots p_{r}\cdot q_{1}\cdot p_{2}\dots q_{s}$, where
$p_{i}\equiv 1$ (mod 4), $q_{j}\equiv -1$ (mod 4) are primes, not
necessarily distinct, then $D(n) \equiv (-1)^{s}(r - s)$ (mod 4).
Hence $m = n' = (-1)^{s_{1}}(r_{1}-s_{1})$ and $n = m' =
(-1)^{s_{2}}(r_{2}-s_{2})$. Because $m$ and $n$ are odd,
$r_{1}-s_{1}$ and $r_{2}-s_{2}$ are not even. However, $n \equiv
(-1)^{s_{1}}$ (mod 4) and $m \equiv (-1)^{s_{2}}$ (mod 4). Hence
$mn \equiv (-1)^{s_{1}}(-1)^{s_{2}}\equiv
(-1)^{s_{1}}(-1)^{s_{2}}(r_{1}-s_{1})(r_{2}-s_{2})$ (mod 4).
Consequently $(r_{1}-s_{1})(r_{2}-s_{2}) \equiv 1$ (mod 4).
\end{proof}


\subsection{Integrals of natural and rational numbers}


Any $a$ such that $a' = b$ is called an integral of $b$. The set
of all such integrals is denoted as $I(b)$. The same number can
have different integrals: $25' = (5^{2})' = 2\cdot5 = 10$ and $21'
= 3\cdot7 = 3 + 7 = 10$. Because $p' = 1$ for any prime, $I(1) =
\mathbb P$. It is shown that 1 is the only natural number with
infinitely many integrals among the natural numbers.

\begin{proposition}\label{prop12}
i) Let $b &lt; 2\cdot3\cdot5\cdots P_{n}$, where $P_{n}$ is the
$n$-th consecutive prime and let $a' = b$, where $a \in \mathbb
N$. Then $a = p_{1}^{n_{1}}\cdots p_{m}^{n_{m}} $, where $m\leq n$
(hence $a$ is divisible by at most $n$ primes $p_{i}$).

ii) If $a' = b &gt; 1$, where $a \in \mathbb N$, then $a \leq
\max(2\cdot3\cdot5\cdots P_{n}, b^{bn})$. Consequently every $b
&gt; 1$ has at most a finite number of integrals $a \in \mathbb N$.
\end{proposition}

\begin{proof} For each natural number $b$ there is a $n \in
\mathbb N$ such that $b &lt; 2\cdot3\cdot5\cdots P_{n}$. If $a =
\prod_{i=1}^{^{m}} p_{i}^{n_{i}}$ has more than $n$ different
prime factors $p_{i}$, then each summand of $a' = (\sum_{i=1}^{m}
\frac {n_{i}}{p_{i}})\cdot b $ has at least $n$ prime factors, and
the smallest of them is not smaller than $2\cdot3\cdot5\cdots
P_{n}$, therefore in that case $a' &gt; 2\cdot3\cdot5\cdots P_{n}&gt;
b$, so it cannot be $a' = b$. Thus $a = p_{1}^{n_{1}}\cdots
p_{m}^{n_{m}}$ and $m \leq n$.

ii) If $a$ contains a factor $p^{p}$, then $b = a'\geq a$, hence $
a \leq b &lt; 2\cdot3\cdot5\cdots P_{n}$. The other possibility is
that all exponents of $a = p_{1}^{n_{1}}\cdots p_{m}^{n_{m}}$ are
smaller than their primes: $n_{i}&lt; p_{i}$. It is necessary to
consider two cases:

If $m = 1$ then $a = p_{1}^{n_{1}}$ and $a' = n_{1}p_{1}^{n_{1}-1}
= \frac{n_{1}a}{q_{1}} = b$. Now $b &gt; 1$ implies $n_{1} \geq 2$
thus $p_{1}$ divides $b$, hence $p_{1}\leq b$ and $a =
\frac{bp_{1}}{n_{1}} \leq bq_{1} \leq b^{2}$.

If $m \geq 2$ then $a = p_{1}^{n_{1}}\cdots p_{m}^{n_{m}} &lt;
p_{1}^{p_{1}}\cdots p_{m}^{p_{m}} &lt; b^{p_{1} + \cdots + p_{m}}
&lt;b^{bn}$ because $p_{i} &lt; a' = b$ for each $p_{i} $.

Hence $a \leq \max(2\cdot3\cdot5\cdots P_{n}, b^{bn})$ and $I(b)
\bigcap \mathbb N$ is finite for any $b &gt; 1$.
\end{proof}

\begin{corollary}\label{coro1} If $a' = p$ and $p$ is a prime, then $a = p_{1}\cdots p_{m}$ and all
$p_{i}&lt; p$. There are some primes without integrals $a \in \mathbb
N$; for example 2,3,17.
\end{corollary}
\begin{proof} If $a = p^{2}c$, then $a' = p\cdot(2 + pc')$ is not a
prime. If $a = p_{1}\cdots p_{m}$, then $p = a' &gt; p_{i}$. Because
$2 &lt; 2\cdot3 $ and $3 &lt; 2\cdot 3$, the only candidates for
integrals of 2 and 3 are numbers of the form $a = p\cdot q$, where
$p,q \in \mathbb P$. However, then $(p\cdot q)' = p + q &gt; 5$,
hence 2 and 3 can have no integrals $a \in \mathbb N$. Ufnarovski
and \AA hlander \cite{Ufna} found their integrals in rational
numbers: $(-\frac{21}{16})' = 2 $, $(-\frac{13}{4})' = 3 $. Any
integral of $17 $ has at most 3 different prime factors because
$17 &lt; 2 \cdot3 \cdot5$. It cannot have only two different prime
factors because if $a = pq$ then $a' = p + q$ and the sum of any
two primes is not 17. However, if $a = pqr$, then $(pqr)' = pq +
pr + qr \geq 2 \cdot 3 + 2 \cdot 5 + 3 \cdot 5 = 31 &gt; 17.$ Thus
there is no integral of 17.
\end{proof}

The numbers $s \in \mathbb N$ not divisible by any square $c^{2}$
where $c &gt; 1$  are called square-free numbers. They can be either
products of different primes or equal to 1. The set of square-free
numbers is denoted $\mathbb S$.

\begin{definition} \emph{Let $a = p_{1}^{n_{1}} \cdots p_{m}^{n_{m}}$, where $p_{i} \in \mathbb P $.
The number $n_{i}$ is called the exponent of a prime $p_{i}$ in
$a$. Let us define the following functions of $a$:}

$s(a)$ \emph{is the greatest square-free divisor of} $a$,
\emph{such that} $\gcd(s(a), \frac {a}{s(a)}) = 1$,

\emph{$p(a) = p_{1}\cdots p_{m} $ is the product of all prime
factors of $a$},

\emph{$f(a) = \frac {a}{p(a)} = \frac {a}{p_{1} \cdots p_{m}}$,
$r(a) = a\cdot p(a) = a\cdot( p_{1}\cdots p_{m})$,}

\emph{$h_{\max}(a) = \max \{n_{1}, \ldots , n_{m}\}$, $h_{\min}(a)
= min \{n_{1}, \ldots ,n_{m}\}$. }

\noindent \emph{For $a = 1$ we define} \emph{$s(a) = p(a) = f(a) =
r(a) = h_{\max}(a) = h_{\min}(a) = 1.$}

\end{definition}

From this definition it follows that if $a \neq s(a)$ then $
h_{min}(\frac {a}{p(a)}) \geq 2.$

\begin{proposition} \label{prop13} i) Let $a \in \mathbb N$. Then $a' = f(a)p(a)a^{*}$, $p(a)a^{*}
\in \mathbb N$ and $a = r(f(a))s(a)$.

ii) If $a \in \mathbb P$ then $p(a)a^{*} = 1 $ and $f(a) = a'.$ If
$a \in \mathbb N$ is not a prime then $p(a)a^{*}
&gt; 1 $ and $f(a) &lt; a'.$

iii) If $a = p_{1}^{n_{1}} \cdots p_{m}^{n_{m}} $ and $n_{i}$ is
not divisible by $p_{i}$ for all $i \in \{1,2,\dots ,m\}$ then
$\gcd(f(a),p(a)a^{*}) = 1$ and $\gcd(a,a') = f(a)$.

iv) If $h_{\min}(a)\geq 2$, then $s(a) = 1$, hence $a = r(f(a))$.
If $h_{\max}(a) = 1$ then $p(a) = a$ and $f(a) = 1$, hence $r(f(a))
= r(1) = 1$.

\end{proposition}

\begin{proof}  i)\, If $a = 1$ then $a' =
a^{*} = 0$ implies $a' = f(a)p(a)a^{*}$. If $a = p_{1}^{n_{1}}
\cdots p_{m}^{n_{m}}$ then $a' = aa^{*} = (\frac
{a}{p(a)})p(a)a^{*}$ and $\frac {a}{p(a)} =  f(a) =
p_{1}^{n_{1}-1} \cdots p_{m}^{n_{m}-1}$. Obviously $p(a)a^{*} =
(p_{1}\cdots p_{m})(\frac {n_{1}}{p_{1}} +\cdots + \frac
{n_{m}}{p_{m}}) \in \mathbb N$. Because $f(a)$ is divisible by
exactly those primes $p_{i}$ for which $p_{i}^{2}$ divides $a$, it
is $r(f(a)) = \frac{a}{s(a)}$. Hence $a = r(f(a))s(a)$.

ii) If $a \in \mathbb P$ then $p(a) = a $, $a' = 1$, $a^{*} =
\frac {1}{a}$, $p(a)a^{*} = a \frac {1}{a} = 1$ and $f(a) = 1.$
Now suppose $a \in \mathbb N$ is not a prime. Then either $a =
p_{1}^{n_{1}}$, where $n_{1} &gt; 1$, or $a = p_{1}^{n_{1}} \cdots
p_{m}^{n_{m}}$, where $ m \geq 2$. In the first case $p(a) =
p_{1}$, $f(a) = p_{1}^{n_{1} -1}$, $a' = n_{1}p_{1}^{n_{1} -1}$,
$a^{*} = \frac {n_{1}}{p_{1}}$, hence $p(a)a^{*} = p_{1}\frac
{n_{1}}{p_{1}} = n_{1} &gt; 1$. In the second case, $p(a)a^{*} =
(p_{1}\cdots p_{m})(\frac {n_{1}}{p_{1}} +\cdots + \frac
{n_{m}}{p_{m}})$ is a sum of at least two natural numbers, hence
$p(a)a^{*} &gt; 1 $. Therefore in both cases $f(a) = \frac
{a'}{p(a)a^{*}} &lt; a'$.

iii) Now $p(a)a^{*} = (p_{1}\cdots p_{m})(\frac {n_{1}}{p_{1}}
+\cdots + \frac {n_{m}}{p_{m}})$ is not divisible by any of the
primes $p_{i} $ dividing $a$ because $n_{i}$ is not divisible by
$p_{i}.$ Hence $p(a)a^{*} = q_{1}^{u_{1}} \cdots q_{t}^{u_{t}} $
and $a' = p_{1}^{n_{1}-1} \cdots p_{m}^{n_{m}-1}q_{1}^{u_{1}}
\cdots q_{t}^{u_{t}} $, where all the primes $p_{1},\cdots ,p_{m},
q_{1},\cdots ,q_{t}$ are distinct. Hence $\gcd(f(a),p(a)a^{*}) = 1
$ and $\gcd(a,a') = f(a)$.

iv) If $h_{\min}(a)\geq 2$, then $p(a) = p(f(a))$, hence $r(f(a))
= a$. If $h_{\max}(a) = 1$ then $p(a) = a $ and  $f(a) = 1$, hence
$ r(f(a)) = 1$. \end{proof}

\begin{definition}If $a = p_{1}^{n_{1}} \cdots p_{m}^{n_{m}} $ where all $n_{i}&lt;
p_{i} $ then $a$ is in the set $\mathbb L $ whose elements are
called "low" numbers.
\end{definition}

Now it is possible to describe the factorization and consequently
obtain some bounds of the integrals of "low" numbers.

\begin{corollary} \label{coro2} i) If $b \in \mathbb L$ then $I(b) \bigcap \mathbb N
\subset \mathbb L$. Moreover, every $a \in I(b) \bigcap \mathbb N$
is of the form $a = r(c)s(a)$, where $b = cd$, $c &lt; b$, $\gcd(c,d)
= 1$, and $b = a' = r'(c)s(a) + r(c)s'(a)$. Therefore:

i.a) If $s(a) = a$ then $r(c) = 1$ hence $c = 1$. Conversely, if
$c = 1$ then $s(a) = a$.

i.b) If $s(a) = 1$ then $a = r(c)$.

i.c) If $r(c) \neq 1 $ then $s(a) \leq \frac{b - r(c)}{r'(c)} \leq
\frac{b}{c}$, hence $a \leq r(b) - r(r(c)) \leq r(b) - 1$.

i.d) If $r(c) \neq 1 $ and $s(a) $ is a prime then $s(a) = \frac{b
- r(c)}{r'(c)}$.

i.e) If $r(c) \neq 1 $ and $s(a) = p_{1}p_{2}$ then $\frac{b -
r(c)}{r'(c) + r(c)} \leq s(a) \leq \frac{b - 5r(c)}{r'(c)}$.

 ii) For every $b = p_{1}^{n_{1}}
\cdots p_{k}^{n_{k}} \in \mathbb L$ there are at most $2^{k} - 1$
different divisors $c$ of $b$ such that there is an integral $a$
of $b$ of the form $a = r(c)s(a)$.

 iii) If $b = p_{1}^{n_{1}} \cdots p_{k}^{n_{k}}$ and $n_{i} = p_{i} - 1 $ for
all $i$, then any integral $a \in I(b) \bigcap \mathbb N$ is a
square-free number: $a = s(a)$.
\end{corollary}
\begin{proof} i) If $a' = b$ and $a$ contains a factor $p_{i}^{p_{i}}$ then $a' = b$ is
also divisible by $p_{i}^{p_{i}}$, too, hence $b$ is not in
$\mathbb L$. Thus, because $b \in \mathbb L$, it must be $a \in
\mathbb L.$ Hence $f(a)$ divides $a' = b$, $f(a) &lt; a'$, and
$r(f(a)) = \frac {a}{s(a)}$. Thus one writes $c = f(a)$, one
really can get all the integrals of $b \in \mathbb L$ in the
described form.

i.a) and i.b) are obvious.

i.c) If $r(c) \neq 1 $ then $r'(c)\neq 0$  and $s(a) = b -
\frac{r(c)s'(a)}{r'(c)} \leq \frac{b - r(c)}{r'(c)} =
\frac{b}{(cp(c))'} = \frac{b}{c'p(c) + cp'(c)} \leq \frac{b}{c}$.
Hence $a = r(c)s(a) \leq c\cdot p(c)\frac{b - r(c)}{c} \leq p(b)b
- p(c)r(c) = r(b) - r(r(c)) \leq r(b) - 1$ because $p(c) =
p(r(c))$.

i.d) If $r(c) \neq 1 $ and $s(a)$ is a prime, then $s'(a) = 1$ and
$s(a) = \frac{b - r(c)}{r'(c)}.$

i.e) If $r(c) \neq 1 $ and $s(a) = p_{1}p_{2}$ then $s'(a) = p_{1}
+ p_{2} \leq 2 + 3 = 5$ hence $ s(a) = \frac{b - 5r(c)}{r'(c)}$.
The inequality $\frac{b - r(c)}{r'(c) + r(c)} \leq s(a)$ follows
from the fact that $p_{1} + p_{2}\leq p_{1}p_{2} $ for any two
primes $p_{1}$ and $p_{2}$ implying $a' = (r(c)p_{1}p_{2})' =
r'(c)p_{1}p_{2} + r(c)(p_{1} + p_{2}) \leq (r'(c) +
r(c))p_{1}p_{2}.$


ii) Because each factor $p_{i}^{n_{i}}$ of $b = cd = p_{1}^{n_{1}}
\cdots p_{k}^{n_{k}}$ belongs either to $c = f(a)$ or $d =
\frac{b}{f(a)}$, and because $c &lt; a'$, there are at most $2^{k} -
1$ possible factors $c$ dividing $b = a'$ such that $a =
r(c)s(a)$. However, different integrals of $b$ may correspond to
the same $c$, as in the example $(3\cdot13)' = (5\cdot11)' = 16$,
in which both integrals $3\cdot13 $ and $5\cdot11$ correspond to
$c = 1$.

iii) If $p_{i} = n_{i} - 1 $ for all $i$, then $r(c) = 1$ or
$r(c)$ contains at least one factor $p_{i}^{p_{i}}$, but then this
is a contradiction because in that case $a = b'$ would also be
divisible by $p_{i}^{p_{i}}$. Hence $r(c) = 1$ and $a = s(a)$.
\end{proof}

\begin{example} Let $b = 5^{2}11$. There are three
possible factorizations $b = cd$ such that $\gcd(c,d)= 1$ and $1
\leq c &lt; b$, corresponding to numbers $c_{1} = 1$, $c_{2} = 5^{2}$
and $c_{3} = 11$. Thus the candidates for $a \in I(b) \bigcap
\mathbb N$ are:

square-free numbers $a = r(1)s(a) = s(a) 2\cdot5^{2}11$, not
divisible by $5$ or $11$,

numbers of the form: $a = r(5^{2})s(a) = 5^{3}s(a)$, where $s(a)
\leq \frac{b}{r'(c)} = \frac{5^{2}11}{3\cdot 5^{2}} = \frac{11}{3}
\leq 4$, hence $s(a) \in \{1,2,3\}$, and

numbers of the form $a = r(11)s(a) = 11^{2}s(a)$, where $s(a) \leq
\frac{b}{r'(c)} = \frac{5^{2}11}{2\cdot 11} = \frac {5^{2}}{2}$,
hence $s(a) \in \{1,2,3,5,6,7,10,11\}$.

It can easily be seen that $s(a) &gt; 1 $ because $s(a) = 1$ implies
$r'(c) = b$, but it is $1' = 0 \neq b$, $(5^{3})' = 3\cdot5^{2}
\neq b$ and $(11^{2})' = 2\cdot11 \neq b$. Thus in the case $c_{2}
= 5^{2}$ it must be $s(a) \in \{2,3\} $ and in the case of $c_{3}
= 11$ it is seen that $s(a) \in \{2,3,5,6,7,10,11\}$ is either a
prime or a product of two primes.

Hence in the case $a = 5^{3}s(a)$ one can use the formula for the
prime $s(a) = \frac{b - r(c)}{r'(c)} = \frac{5^{2}11 -
5^{3}}{3\cdot5^{2}} = \frac{11 - 5} {3} = 2$ and one can verify
directly that $(2\cdot5^{3})' = 5^{3} + 2\cdot3\cdot5^{2} =
5^{2}(5 + 6) = b$.

Now let $a = 11^{2}s(a)$. If $s(a)$ is a prime one gets $s(a) =
\frac{b - r(c)}{r'(c)} = \frac{5^{2}11 - 11^{2}}{2\cdot 11)} =
\frac {25 - 11}{2} = 7$ and one can check directly that
$(11^{2}\cdot7)' = 2\cdot11\cdot7 + 11^{2}  = 11(14 + 11 ) = b$.
It can easily be seen that $s(a)$ cannot be a product of two
primes because that would imply a contradiction: $0 \leq s(a) \leq
\frac{b - 5r(c)}{r'(c)} = \frac{5^{2}11 -5\cdot11^{2}}{2\cdot11} &lt;
0$. \end{example}


Square-free integrals corresponding to $c_{1} = 1$ are more
difficult to find. It is necessary to check the derivatives of all
square-free numbers $a \leq \frac{b^{2}}{4}$ and compare them with
$b$.

A similar estimate as in Proposition~\ref{prop10} can be made
about positive solutions to the equation $(\frac{a}{b})' =
\frac{2}{p}$, where $p \in \mathbb P $.

\begin{definition} Let $P_{i+1}$ denote the $i$-th odd prime.
For any $d \in \mathbb R$ let $O_{d} = 1\cdot\prod_{i=1}^{m}
P_{i+1}$ denote the product of the first $m$ odd primes such that
the sum of their reciprocals, denoted $R(d)$, is not smaller than
$d$.
\end{definition}

Thus $R(d) = \sum_{i=1}^{m} \frac {1}{P_{i+1}} \geq d$. Because
the series $\sum_{i=1}^{\infty}\frac{1}{P_{i+1}}$ diverges,
$O_{d}$ is well defined for any real $d$.

\begin{example} If $d \leq \frac{1}{3}$ then $O_{d}
= P_{2} = 3.$ If $\frac{1}{3} \leq d \leq \frac{1}{3} +
\frac{1}{5}$ then  $O_{d} = P_{2}P_{3} = 3\cdot5 = 15.$ If
$\frac{1}{3} + \frac{1}{5}\leq d \leq \frac{1}{3} + \frac{1}{5} +
\frac{1}{7}$ then  $O_{d} = P_{2}P_{3}P_{4} = 3\cdot5\cdot7 =
105.$ \end{example}

\begin{proposition}\label{prop14} Let $(\frac{a}{b})' = \frac{2}{p}$,
where $p \in \mathbb P$, $p&gt; 2$, $\gcd(a,b) = 1$, $\frac{a}{b} &gt;
0$. Then $a = p_{1}\cdots p_{m}$ is an odd square-free number with
$m \geq 9$ prime factors $p_{i} \in \mathbb P$ and $b$ is of the
form $b = q_{1}^{q_{1}n_{1}} \dots q_{s}^{q_{s}n_{s}}$ where
$q_{i} \in \mathbb P $ and $n_{i} \in \mathbb N $. Moreover, $L(b)
\in \mathbb N$, $L(a) &gt; L(b) \geq 1$ and $a'
&gt; a &gt; O_{1} \geq 3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29
$.  \end{proposition}

\begin{proof} $(\frac{a}{b})' = \frac{2}{p}$ implies $\frac{a'b -
ab'}{b^{2}}  = \frac{2}{p}$, hence $b = pc$, where $c \in \mathbb
N$. Therefore $a'pc - ab' = 2c^{2}p$ and because $a$ and $b$ have
no common prime factors, we have $b' = kc$. Hence $a'pc - akc =
2c^{2}p$ and by dividing this equation by $c$ one gets $a'p - ak =
2cp,$ hence $k = pd$ because $p$ divides $b$ so it cannot divide
$a$. This implies $a'p - apd = 2cp$, hence $a' - ad = 2c$, and $b'
= pdc = bd$. Thus $d = L(b) = b^{*}$. By Proposition~\ref{prop2},
 $b = q_{1}^{q_{1}n_{1}} \cdots q_{s}^{q_{s}n_{s}} $, where $n_{1}
+ \cdots + n_{s} = d.$

Because $\frac{a}{b} &gt; 0$, it can be assumed $a &gt; 0$ and $b &gt; 0$.
It must be $a &gt; 1$ because $a = 1$ implies $a' = 0$ and $-d = 2c$,
implying a contradiction: $0 = 2c + d &gt; 0$.

From the equation $a' - ad = 2c$ follows $\gcd(a,a') = 1$, because
any such common prime factor different from 2 would also divide
$c$ and $b$ and this would contradict $\gcd(a,b) = 1$, and if 2
divides $a$ and $a'$ then it must be $a = 4e$, hence $a' = 4(e +
e') $ and $4(e' + e - 4 ed) = 2c$ would imply that 2 also divides
$b$.

Now it can be seen that $a$ cannot be divisible by $2$. In that
case $2$ would also divide $a'$ or $p$. However, this is
impossible because $\gcd(a',a) = 1$ and $p &gt; 2$.

Because $\gcd(a,a') = 1$, $a$ must be a square-free number. Because
$a
&gt; 1$, it must be: $a = p_{1}\cdots
p_{m}$, and because $ a' - ad = 2c &gt; 0$ it is $a' &gt; ad \geq a $,
hence $L(a) = \frac{a'}{a} = \sum_{i=1}^{m}\frac{1}{p_{i}} &gt; d
\geq 1$. Therefore $a &gt; O_{d} \geq O_{1} \geq 3\cdot 5 \cdot 7
\cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29$ and $ m
\geq 9$, because the sum of the reciprocals of the first eight odd
primes is less than 1: $\frac{1}{3} + \frac{1}{5} + \frac{1}{7} +
\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19} +
\frac{1}{23} = 0.9987\cdots &lt; 1$.
\end{proof}

Ufnarovski and \AA hlander \cite{Ufna} conjectured that some
rational numbers have no integrals. The above proposition shows
that the integrals of $\frac{2}{3}$ are not easy to find, if they
exist at all. The same holds for any $\frac{2}{p}, p \in \mathbb
P.$

In Proposition~\ref{prop4}.iii) I showed that if $y \in N_{y^{*}}
= N_{r+r^{*}}$ and if $r + r^{*} \neq 0$ then there is exactly one
$x \in N_{r}$ such that $x' = y$ and this is $x = \frac{y}{r}$.
Hence to find a rational integral $x$ of any nonzero rational
number $y$ one just has to find a rational number $r = x^{*}$ such
that $r + r^{*} = y^{*}$. In other words, one has to find a
logarithmic class $N_{r}$ such that $D(N_{r}) = N_{y^{*}}$. The
equation $r + r^{*} = y^{*}$ translates into $r^{2} - y^{*}r + r'
= 0$. Perhaps for some rational numbers $y$ this equation cannot
be satisfied by any rational number $r$, hence such $y$ cannot
have a rational integral $x$. Expressing $r$ as a function of its
derivative $r'$, one gets at most two different solutions $r_{1,2}
= \frac{y^{*} \pm \sqrt{(y^{*})^{2} - 4r'}}{2}$. Now $ r \in
\mathbb Q$ implies $(y^{*})^{2} - 4r' \geq 0$ and $r' \leq
\frac{(y^{*})^{2}}{4}.$ Because $x \in \mathbb Q$ implies $r =
x^{*} \in \mathbb Q$ the expression $(y^{*})^{2} - 4r' = q^{2}$
must be a square of a rational number $q$.

\begin{example} If $y = \frac{2}{3}$ then $y^{*} =
\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$. If there is a positive
rational number $x = \frac{a}{b} \in N_{r}$ such that $ x' = y =
\frac{2}{3}$ then $r = x^{*} = (\frac{a}{b})^{*} = \frac{y}{x} =
\frac{2}{3x}&gt; 0$ satisfies the equation $r + r^{*} = y^{*} =
\frac{1}{6}$. This equation translates into $r^{2} - \frac{1}{6}r
+ r' = 0$. For each $r'$ one gets at most two different solutions
$r_{1,2} = \frac{y^{*} \pm \sqrt{(y^{*})^{2} - 4r'}}{2} =
\frac{\frac{1}{6} \pm \sqrt{\frac{1}{36} - 4r'}}{2} = \frac{1}{12}
\pm \sqrt{\frac{1}{144} - r'}$. Hence $r' \leq
\frac{(y^{*})^{2}}{4} = \frac{1}{144}$.

It is known from Proposition~\ref{prop14} that $r = a^{*} - b^{*}
= \sum_{i=1}^{m}\frac{1}{p_{i}} - d &gt; 0$, where $ d \in \mathbb
N$. Thus $x = \frac{y}{r} &gt; 0$. Now $r' = (\frac{2}{3x})' =
\frac{3x - 6}{9x^{2}} = \frac{x - 2}{3x^{2}} = \frac{x -
2}{3x^{2}}\frac{3x}{2}\frac{2}{3x}$, hence $r^{*} = \frac{x -
2}{2x}$. If $x \leq 2$ then $r^{*} \leq 0 $ and $r = y^{*} - r^{*}
\leq y^{*} = \frac{1}{6} $ leads to a contradiction: $x =
\frac{y}{r} \geq \frac{y}{y^{*}} = \frac{\frac{2}{3}}{\frac{1}{6}}
= 4$. Thus it must be $x &gt; 2$, hence $r^{*} &gt; 0$. Now $0 &lt; r =
y^{*} - r^{*} &lt; y^{*}  = \frac{1}{6} $ implies $x = \frac{y}{r}
&gt; \frac{y}{y^{*}} = \frac{\frac{2}{3}}{\frac{1}{6}} = 4$, hence $a
&gt; 4b$. Thus there is no positive rational number $x$ such that $x \leq 4$ and $x'
= \frac{2}{3}$.\end{example}

\subsection{Partial derivatives and partial differential equations}

\begin{definition} \emph{Let $a = \prod_{i=1}^{k}p_{i}^{\, x_{i}}$ be the
factorization of $a \in \mathbb N$ into primes. The partial
derivative $\frac{\partial a} {\partial pi} = D_{p_{i}}(a)$ is
defined as $\frac{\partial a} {\partial p_{i}} =
a\frac{x_{i}}{p_{i}}.$ If $p \in \mathbb P$ is not in the
factorization of $a$, then $\frac{\partial a} {\partial p_{i}} =
0.$}
\end{definition}

From this definition it immediately follows that $D(a) =
a\sum_{i=1}^{k}\frac{\partial a} {\partial p_{i}}. $ One can also
 define higher partial derivatives; for example:
$\frac{\partial ^{2} a}{\partial p_{i}\partial p_{j}} =
D^{2}_{p_{i}p_{j}} = $ $\frac{\partial (\partial a/\partial
p_{j})}{\partial p_{i}} $, $\frac{\partial ^{2} a}{\partial
p_{i}\partial p_{j}} = D^{2}_{p_{i}^{2}} = $ $\frac{\partial
(\partial a/\partial p_{i})}{\partial p_{i}} $, etc., and study
partial differential equations.

\begin{example}
$D_{2}(2^{3}\cdot5^{4}) = 3\cdot2^{2}\cdot5^{4} $,
$D_{5}(3\cdot2^{2}5^{4} )  = 3\cdot2^{2}\cdot4\cdot5^{3} =
3\cdot2^{4}\cdot5^{3}  $

\noindent $D_{5}(2^{3}\cdot5^{4} ) = 2^{3}\cdot4\cdot5^{3} =
2^{5}\cdot5^{3}$ and $D_{2}(2^{5}\cdot5^{3} ) =
5\cdot2^{4}\cdot5^{3}$. Thus the order of applying $D_{p} $ and
$D_{q}$ is important: $D_{p}D_{q}$ is not always equal to
$D_{q}D_{p} $. \end{example}

\begin{proposition}\label{prop15} If $a = p^{e}q^{f} $ and $\gcd(e,q) = 1$ and $ \gcd(f,p) =
1$, then $D_{p}D_{q}(a) = D_{q}D_{p}(a). $
\end{proposition}
\begin{proof} In that case it is $D_{p}D_{q}(p^{e}q^{f}) =
D_{p}(p^{e}fq^{f-1}) = ep^{e-1}fq^{f-1}$ and
$D_{q}D_{p}(p^{e}q^{f} ) = D_{q}(ep^{e-1}q^{f} ) =
ep^{e-1}fq^{f-1}$.
\end{proof}

\begin{proposition}\label{prop16} Let $n = p^{e}q^{f}c$, where $p,q$ are primes not dividing $c$.
 Then $D_{p}(p^{e}q^{f}c) = D_{q}(p^{e}q^{f}c)  $ if and
only if $e = kp $ and $f = kq$, where $ k \in \mathbb N$.
\end{proposition}

\begin{proof} From $D_{p}(p^{e}q^{f}c)= ep^{e-1}q^{f}c =
fp^{e}q^{f-1}c = D_{q}(p^{e}q^{f}c)  $ follows $eq = bp$, hence $e
= kp$ and $f = kq$. Then $D_{p}(p^{kp}q^{kq}c) = k(p^{kp}q^{kq}c)
= D_{q}(p^{e}q^{f}c).$ \end{proof}

\begin{proposition}\label{prop17} i) The only solutions to the partial differential
equation $D^{2}_{p^{2}}(n) = n$ in natural numbers are $n =
p^{pk}c$, where $\gcd(c,p) = 1$.

ii) If $n = p^{e}d$ where $\gcd(p,d) = 1$ and $e \geq p$ then
$D^{2}_{p^{2}}(n)\geq n.$

 iii) The only solutions to the
partial differential equation $(D^{2}_{p^{2}} + D^{2}_{q^{2}})(n)
= n$ in natural numbers are $n = p^{p}c$ and $n = q^{q}c$, where
$\gcd(pq,c) = 1$.
\end{proposition}
\begin{proof} i) Let $n = p^{e}c$, where $e \in \mathbb N \bigcup \{0\}$ and
$c \in \mathbb N$ is not divisible by $p$.

If $e = 0$ then $D_{p}(n) = 0$, hence $D^{2}_{p^{2}}(n) = 0$. If
$e = kp$ and $k \in \mathbb N$ then $D_{p}(n) = D_{p}(p^{pk}c) =
pkp^{pk-1}c = p^{pk}c = n$ hence $D^{2}_{p^{2}}(n) = n$. If $e =
1$ then $D_{p}(n) = c$, hence $D^{2}_{p^{2}}(n) = D_{p}(c) = 0$.

ii) If $e = kp$ and $k \in \mathbb N$ then $D^{2}_{p^{2}}(n) = n$,
as is already known from i). If $\gcd(e,p) = 1$ and $e &gt; p$ then
$D_{p}(n) = D_{p}(p^{e}c) = ep^{e-1}c$, hence $D^{2}_{p^{2}}(n) =
D_{p}(ep^{e-1}c) = e(e - 1)p^{e-2}c \geq (p + 1)pp^{e-2}c &gt; p^{e}c
= n$.

iii). One can write $n = p^{e}q^{f}c $, where $\{e,f\} \subseteq
\mathbb N \bigcup \{0\}$ and $c \in \mathbb N$ is not divisible by
$p$ or $q$. So it is necessary to study the equation
$(D^{2}_{p^{2}} + D^{2}_{q^{2}})( p^{e}q^{f}c) = p^{e}q^{f}c. $ If
$f = 1$ then $D^{2}_{q^{2}}(p^{e}c) = 0$ and one gets the equation
$D^{2}_{p}(n) = n$ whose only solution is $n = p^{p}c $, as is
known from i). Likewise if $e = 1$ one gets $n = q^{q}c. $

In the case of $e \geq 2$ and $f \geq 2$, one can use the
following argument:

Obviously $\gcd(qc,p) = 1$ implies that $D^{2}_{p^{2}}(n)$ is
divisible by $q^{f}c$. Likewise $\gcd(pc,q) = 1$ implies that
$D^{2}_{q^{2}}(n)$ is divisible by $p^{e}c$. So one can write
$D^{2}_{p^{2}}(n) = aq^{f}c$, $D^{2}_{q^{2}}(n) = bp^{e}c$, where
$a,b \in \mathbb N \bigcup \{0\}$. Suppose $a \neq 0$ and $b \neq
0$. Hence one gets the equation $aq^{f}c + bp^{e}c = p^{e}q^{f}c$.
This equation can be solved only if $a$ is divisible by $p^{e}$
and if $b$ is divisible by $q^{f}$. Suppose $a \neq 0$ and $b \neq
0$. Then $D^{2}_{p^{2}}(n) \geq n$ and $D^{2}_{q^{2}}(n) \geq n$,
hence $D^{2}_{p^{2}}(n) + D^{2}_{q^{2}}(n)\geq 2n$ and the
equality is not possible. Hence it must be either $a = 0$ or $b =
0$. However, this is not possible if $e \geq 2$ and $f \geq 2$.
\end{proof}


\section{Concluding remarks}

I have proved some new results about the arithmetic derivative and
integral. I have defined arithmetic partial derivatives and solved
some arithmetic partial differential equations. I have shown that,
for any solution to the system $m' = n, n' = m$ in natural
numbers, at least one of the numbers $m$ and $n$ is not smaller
than the number $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot
17 \cdot 19 \cdot 23 \cdot 31 \cdot 37 \cdot41 \cdot 43 \cdot 47
\cdot 53 \cdot 59$. The arithmetic derivative can be defined on
sequences of numbers as follows: $D(a_{1},a_{2}, \ldots , a_{n},
\ldots ) = (D(a_{1}),D(a_{2}), \ldots , D(a_{n}), \ldots ) ).$
Thus taking any integer sequence $(a) = (a_{1},a_{2}, \ldots ,
a_{n}, \ldots )$ one can get an infinite family of derived
sequences $D(a), D^{2}(a), \ldots, D^{k}(a), \ldots .$ I believe
many other useful applications of the arithmetic derivative will
be discovered.

\section{Acknowledgements}
I thank the referee for many suggestions that
helped me to improve this paper. I am also grateful to Marko
Petkov\v{s}ek for his explanation of why there are uncountably
many functions on the unit circle in the complex plane satisfying
the Leibnitz rule (Proposition 1.ii).


\begin{thebibliography}{9}

\bibitem{Barb} E. J. Barbeau, Remarks on arithmetic derivative,
\emph{Canad. Math. Bull}, \textbf{4} (1961), 117--122.

\bibitem{Stay} M. Stay, Generalized number derivatives, \emph{J.
Integer Seq.}, \textbf{8} (2005), 
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Stay/stay44.html}{Article 05.1.4}.

\bibitem{Ufna} V. Ufnarovski and B. \AA hlander, How to differentiate
a number, \emph{J. Integer Seq.}, \textbf{6} (2003), 
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html}{Article 03.3.4}.

\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11A25; Secondary 11A41, 11N05, 11N56, 11Y55.

\noindent \emph{Keywords: } 
arithmetic derivative, arithmetic
antiderivative, arithmetic partial derivative, Leibnitz rule.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000040},
\seqnum{A000290},
\seqnum{A051674}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received May 19 2011;
revised version received  March 18 2012.
Published in {\it Journal of Integer Sequences}, March 25 2012.

\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>