5,314 research outputs found

### How to obtain the continued fraction convergents of the number $e$ by neglecting integrals

In this note, we show that any continued fraction convergent of the number $e
= 2.71828...$ can be derived by approximating some integral $I_{n, m} :=
\int_{0}^{1} x^n (1 - x)^m e^x d x$ $(n, m \in \mathbb{N})$ by 0. In addition,
we present a new way for finding again the well-known regular continued
fraction expansion of $e$.Comment: 7 pages, To appea

### On the derivatives of the integer-valued polynomials

In this paper, we study the derivatives of an integer-valued polynomial of a
given degree. Denoting by $E_n$ the set of the integer-valued polynomials with
degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying
the property: $\forall P \in E_n, c_n P' \in E_n$ is $c_n = \mathrm{lcm}(1 , 2
, \dots , n)$. As an application, we deduce an easy proof of the well-known
inequality $\mathrm{lcm}(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq
1$). In the second part of the paper, we generalize our result for the
derivative of a given order $k$ and then we give two divisibility properties
for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on
this study, we conclude the paper by determining, for a given natural number
$n$, the smallest positive integer $\lambda_n$ satisfying the property:
$\forall P \in E_n$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in E_n$.
In particular, we show that: $\lambda_n = \prod_{p \text{ prime}}
p^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$).Comment: 17 page

### Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

In this paper, we present new structures and results on the set \M_\D of
mean functions on a given symmetric domain \D of $\mathbb{R}^2$. First, we
construct on \M_\D a structure of abelian group in which the neutral element
is simply the {\it Arithmetic} mean; then we study some symmetries in that
group. Next, we construct on \M_\D a structure of metric space under which
\M_\D is nothing else the closed ball with center the {\it Arithmetic} mean
and radius 1/2. We show in particular that the {\it Geometric} and {\it
Harmonic} means lie in the border of \M_\D. Finally, we give two important
theorems generalizing the construction of the \AGM mean. Roughly speaking,
those theorems show that for any two given means $M_1$ and $M_2$, which satisfy
some regular conditions, there exists a unique mean $M$ satisfying the
functional equation: $M(M_1, M_2) = M$.Comment: 23 pages. To appea

### An analog of the arithmetic triangle obtained by replacing the products by the least common multiples

In this paper, we introduce an analog of the Al-Karaji arithmetic triangle by
substituting in the formula of the binomial coefficients the products by the
least common multiples. Then, we give some properties and some open questions
related to the obtained triangle.Comment: 10 page

### Summation of certain infinite Fibonacci related series

In this paper, we find the closed sums of certain type of Fibonacci related
convergent series. In particular, we generalize some results already obtained
by Brousseau, Popov, Rabinowitz and others.Comment: 14 page

### A curious result related to Kempner's series

It is well known since A. J. Kempner's work that the series of the
reciprocals of the positive integers whose the decimal representation does not
contain any digit 9, is convergent. This result was extended by F. Irwin and
others to deal with the series of the reciprocals of the positive integers
whose the decimal representation contains only a limited quantity of each digit
of a given nonempty set of digits. Actually, such series are known to be all
convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of
the reciprocal of the positive integers whose the decimal representation
contains the digit 9 exactly $r$ times, the impressive obtained result is that
$S^{(r)}$ tends to $10 \log{10}$ as $r$ tends to infinity!Comment: 5 pages, to appear in (The) American Mathematical Monthl

### An explicit formula generating the non-Fibonacci numbers

We show among others that the formula: $\lfloor n +
\log_{\Phi}\{\sqrt{5}(\log_{\Phi}(\sqrt{5}n) + n) -5 + \frac{3}{n}\} - 2
\rfloor (n \geq 2),$ (where $\Phi$ denotes the golden ratio and $\lfloor
\rfloor$ denotes the integer part) generates the non-Fibonacci numbers.Comment: 5 page

### A study of a curious arithmetic function

In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to
\mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in
\mathbb{N}$, $\ell$ odd). We show several important properties about that
function and then we use them to obtain some curious results involving the
2-adic valuation.Comment: To appea

### A measure of intelligence of an approximation of a real number in a given model

In this paper, we present a way to measure the intelligence (or the interest)
of an approximation of a given real number in a given model of approximation.
Basing on the idea of the complexity of a number, defined as the number of its
digits, we introduce a function noted $\mu$ (called a measure of intelligence)
associating to any approximation $\mathbf{app}$ of a given real number in a
given model a positive number $\mu(\mathbf{app})$, which characterises the
intelligence of that approximation. Precisely, the approximation $\mathbf{app}$
is intelligent if and only if $\mu(\mathbf{app}) \geq 1$. We illustrate our
theory by several numerical examples and also by applying it to the rational
model. In such case, we show that it is coherent with the classical rational
diophantine approximation. We end the paper by proposing an open problem which
asks if any real number can be intelligently approximated in a given model for
which it is a limit point.Comment: 22 page

### Results and conjectures related to a conjecture of Erd\H{o}s concerning primitive sequences

A strictly increasing sequence $\mathscr{A}$ of positive integers is said to
be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed
that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where
$\mathscr{A}$ is a primitive sequence different from $\{1\}$, are all
convergent and their sums are bounded above by an absolute constant. Besides,
he conjectured that the upper bound of the preceding sums is reached when
$\mathscr{A}$ is the sequence of the prime numbers. The purpose of this paper
is to study the Erd\H{o}s conjecture. In the first part of the paper, we give
two significant conjectures which are equivalent to that of Erd\H{o}s and in
the second one, we study the series of the form $\sum_{a \in \mathscr{A}}
\frac{1}{a (\log a + x)}$, where $x$ is a fixed non-negative real number and
$\mathscr{A}$ is a primitive sequence different from $\{1\}$. In particular, we
prove that the analogue of Erd\H{o}s's conjecture for those series does not
hold, at least for $x \geq 363$. At the end of the paper, we propose a more
general conjecture than that of Erd\H{o}s, which concerns the preceding series,
and we conclude by raising some open questions.Comment: 11 page

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