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Bakir Farhi

Publications and source records attributed to Bakir Farhi.

At least 37 records · Page 2Linked to original sources
arXiv2020

Nontrivial effective lower bounds for the least common multiple of some quadratic sequences

This paper is devoted to studying the numbers $L_{c,m,n} := \mathrm{lcm}\{m^2+c ,(m+1)^2+c , \dots , n^2+c\}$, where $c,m,n$ are positive integers such that $m \leq n$. Precisely, we prove that $L_{c,m,n}$ is a multiple of the rational number \[\frac{\displaystyle\prod_{k=m}^{n}\left(k^2+c\right)}{c \cdot (n-m)!\displaystyle\prod_{k=1}^{n-m}\left(k^2+4c\right)} ,\] and we derive (as consequences) some nontrivial lower bounds for $L_{c,m,n}$. We prove for example that if $n- \frac{1}{2} n^{2/3} \leq m \leq n$, then we have $L_{c,m,n} \geq λ(c) \cdot n e^{3 (n - m)}$, where $λ(c) := \frac{e^{- \frac{2 π^2}{3} c - \frac{5}{12}}}{(2 π)^{3/2} c}$. Further, it must be noted that our approach (focusing on commutative algebra) is new and different from those using previously by Farhi, Oon and Hong.

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arXiv2019

Summation of certain infinite Lucas-related series

In this paper, we find the sums in closed form of certain type of Lucas-related convergent series. More precisely, we generalize the results already obtained by the author in his arXiv paper entitled: "Summation of certain infinite Fibonacci related series".

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arXiv2018

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 $λ_n$ satisfying the property: $\forall P \in E_n$, $\forall k \in \mathbb{N}$: $λ_n P^{(k)} \in E_n$. In particular, we show that: $λ_n = \prod_{p \text{ prime}} p^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$).

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arXiv2017

Results and conjectures related to a conjecture of Erdős 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. Erdős 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 Erdős conjecture. In the first part of the paper, we give two significant conjectures which are equivalent to that of Erdős 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 Erdős'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 Erdős, which concerns the preceding series, and we conclude by raising some open questions.

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arXiv2015

On the possible quantities of Fibonacci numbers that occur in some type of intervals

In this paper, we show that for any integer $a \geq 2$, each of the intervals $[a^k , a^{k + 1})$ ($k \in \mathbb{N}$) contains either $\left\lfloor \frac{\log a}{\logΦ}\right\rfloor$ or $\left\lceil \frac{\log a}{\logΦ}\right\rceil$ Fibonacci numbers. In addition, the density (in $\mathbb{N}$) of the set of the all natural numbers $k$ for which the interval $[a^k , a^{k + 1})$ contains exactly $\left\lfloor \frac{\log a}{\logΦ}\right\rfloor$ Fibonacci numbers is equal to $\left(1 - \left\langle \frac{\log a}{\logΦ}\right\rangle\right)$ and the density of the set of the all natural numbers $k$ for which the interval $[a^k , a^{k + 1})$ contains exactly $\left\lceil \frac{\log a}{\logΦ}\right\rceil$ Fibonacci numbers is equal to $\left\langle \frac{\log a}{\logΦ}\right\rangle$.

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arXiv2013

A curious formula related to the Euler Gamma function

In this note, we prove that for all $x \in (0 , 1)$, we have: $$ \logΓ(x) = \frac{1}{2} \logπ+ π\boldsymbolη \left(\frac{1}{2} - x\right) - \frac{1}{2} \log\sin(πx) + \frac{1}π \sum_{n = 1}^{\infty} \frac{\log n}{n} \sin(2 πn x) , $$ where $Γ$ denotes the Euler Gamma function and $$ \boldsymbolη := 2 \int_{0}^{1} \logΓ(x) \cdot \sin(2 πx) \, d x = 0.7687478924\dots $$

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arXiv2011

An explicit formula generating the non-Fibonacci numbers

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

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arXiv2010

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.

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arXiv2010

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: $M(M_1, M_2) = M$.

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arXiv2009

On the series of the reciprocals lcm's of sequences of positive integers: A curious interpretation

In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $τ_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to + \infty} \frac{1}{N} \sum_{n = 1}^{N} τ_n = \sum_{n = 0}^{\infty} \frac{1}{\lcm\{a \in \A | a \leq n\}} .$${quote} In the two particular cases when $\A$ is the set of all positive integers and when $\A$ is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of ${(τ_n)}_n$. Furthermore, we discuss the irrationality of the limit of $τ_n$ (in the average sense) by applying a result of Erdős.

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arXiv2009

Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis

Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ in function of the order $h$ of $A$ and some parameters related to $X$ and $A$. If the parameter in question is the cardinality of $X$, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in $h$ with degree $(|X| + 1)$. Here, by taking instead of the cardinality of $X$ the parameter defined by $d := \frac{\diam(X)}{\gcd\{x - y | x, y \in X\}}$, we show that the order of $A \setminus X$ is bounded above by $(\frac{h (h + 3)}{2} + d \frac{h (h - 1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmetic progression of length $\geq 3$, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both $X$ and $A$, we get upper bounds which are polynomials in $h$ with degree only 2.

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arXiv2008

New results on the least common multiple of consecutive integers

When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k \in \mathbb{N})$, defined by $g_k(n) := \frac{n (n + 1) ... (n + k)}{\lcm(n, n + 1, >..., n + k)}$ $(\forall n \in \mathbb{N} \setminus \{0\})$. He proved that $g_k$ $(k \in \mathbb{N})$ is periodic and $k!$ is a period of $g_k$. He raised the open problem consisting to determine the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang have improved the period $k!$ of $g_k$ to $\lcm(1, 2, ..., k)$. In addition, they have conjectured that $P_k$ is always a multiple of the positive integer $\frac{\lcm(1, 2, >..., k, k + 1)}{k + 1}$. An immediate consequence of this conjecture states that if $(k + 1)$ is prime then the exact period of $g_k$ is precisely equal to $\lcm(1, 2, ..., k)$. In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $P_k$ $(k \in \mathbb{N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $\lcm(1, 2, ..., k)$ not divisible by some prime.

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