How to obtain the continued fraction convergents of the number ee 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