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M. BOUDAOUD Abdelmadjid

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Department

Mathematics Department

Research Interests

Specialized in Mathematics Department. Focused on academic and scientific development.

Contact Info

University of M'Sila, Algeria

Email : abdelmadjid.boudaoud@univ-msila.dz

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Recent Publications

2025-06-17

Infinite Number of Changes of Sign for A Difference of Two Number-Theoretic Functions

n this paper, wepresenttwonumber-theoreticfunctionsF andGforwhichprF (n)−pr−1G(n)
is both positive andnegativeinfinitely often, where nhasatleastk distinct primefactors (k ≥ 1)
and (pr−1,pr) is a couple of two consecutive primes. To be precise, we will construct infinite
sequences (ni)i≥1
,(mi)i≥1
such that,
F (ni)
G(ni) > pr−1
pr
> F(mi)
G(mi) , for i = 1,2,...,
where each ni and mi has k distinct prime factors and F (t) and G(t) are either the Kernel or
the Euler’s function of the positive integer t.

Citation

M. BOUDAOUD Abdelmadjid, Bellaouar Djamel, , (2025-06-17), "Infinite Number of Changes of Sign for A Difference of Two Number-Theoretic Functions", [national] Malaysian Journal of Mathematical Sciences , Malaysian Journal of Mathematical Sciences (MJMS) Institute for Mathematical Research Universiti Putra Malaysia 43400 UPM Serdang Selangor, MALAYSIA.

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2024-10-29

Solutions of the equation d(kn) =φ(φ(n))

Abstract:Letd(n)andφ(n)denote the number of positive integers dividing the positive integernand the Euler’s phi functionrepresenting the numbers less than and prime ton, respectively. In this paper, we determine all solutions ofthe equationd(n) =φ(φ(n))and we prove that the equationd(kn) =φ(φ(n))has a finite number of solutions for anyk≥1. Further, we characterize all solutionsof the last equation whenkis prime.Keywords:Arithmetic functions; diophantine equations; primes; factorization.2020 Mathematics Subject Classification. 11A25, 11D72, 11A41, 11Y05.

Citation

M. BOUDAOUD Abdelmadjid, (2024-10-29), "Solutions of the equation d(kn) =φ(φ(n))", [national] Jordan Journal of Mathematics and Statistics.Yarmouk UniversityDOI:https://doi.org/10.47013/17.3.1Solutions of the equationd(kn) =φ(φ(n))Amroune Zahra1,∗, Bellaouar Djamel2and Boudaoud Abdelmadjid31Laboratory of Pure and Applied Mathematics (LMPA), Univer , Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan.

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2024-09-17

Some representations of unlimited natural numbers

Abstract. Based on the work of K. Hrbáµcek [13], we prove that every unlimited natural
number! isof theform! = !1 !2+!3 !4 inatleastk di¤erentways(k 1islimited), where
!i 2 N is unlimited and !i=!j is appreciable for 1 i;j 4. Other similar representations
of unlimited natural numbers are also presented.
2010 Mathematics Subject Classi cation 26E35 (primary); 03H05, 03H15, 11A51,
11U10 (secondary)
Keywords: factoring of integers, nonstandard analysis, unlimited integers.

Citation

M. BOUDAOUD Abdelmadjid, (2024-09-17), "Some representations of unlimited natural numbers", [international] ICAA'2024 , BARIKA, Algeria

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2023-10-01

Notes on the equation d(n)=d(phi(n)) and related inequalities

Abstract. Let d(n) denote the number of positive integers dividing the positive integer n, and
let ’ (n) denote Euler’s function representing the number of numbers less than and prime to n.
In this paper, we present some notes on the equation d (n) = d (’ (n)). Several results on the
related inequalities are also obtained

Citation

M. BOUDAOUD Abdelmadjid, (2023-10-01), "Notes on the equation d(n)=d(phi(n)) and related inequalities", [national] Mathematcia Slovaca , Mathematical Institute, Slovak Academy of Sciences (SLOVAKIA)

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2023-04-26

A class of solutions of the equation d (n2) = d (φ (n))

For any positive integer n let d (n) and φ (n) be the number of divisors of n and the Euler’s phi function of n, respectively. In this paper we present some notes on the equation
d (n.n) = d (φ (n)). In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson’s conjecture implies that d (n.n) = d (φ (n))
infnitely often.
Keywords: Diophantine equations, Euler’s phi function, Divisor function.
2010 Mathematics Subject Classifcation: 11A25, 11A41, 11D99.
Copyright © 2023 by the Authors. This is an Open A

Citation

M. BOUDAOUD Abdelmadjid, Bellaouar Djamel, Department of Mathematics, University 08 Mai 1945 Guelma., zahra.amroune@univ-msila.dz, , (2023-04-26), "A class of solutions of the equation d (n2) = d (φ (n))", [national] Notes on Number Theory and Discrete Mathematics , Publishing House of Bulgarian Academy of Sciences.

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2022

Représentation des entiers et formes quadratiques.

C'est une conférence sur les représentations des entiers et les formes quadratiques.

Citation

M. BOUDAOUD Abdelmadjid, (2022), "Représentation des entiers et formes quadratiques.", [national] Rencontre d’Analyse Mathématiqueet ses Applications (RAMA) Université de M’Sila, le 26 octobre 2022 , M'sila

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2020

Representation of Integers: A nonclassical point of view

Abstract: In [2], A. Boudaoud asked the following question: Which n 2 N
unlimited can be represented in the form n = s + !1!2, where s 2 Z is limited
and !1, !2 2 N are unlimited? In this paper we partially answer this question, ie
we present some families of unlimited positive integers which can be written as the
sum of a limited integer and the product of at least two unlimited positive integers.
2010 Mathematics Subject Classification 26E35,03H05 (primary); 11A51, 11A41,
11B83. (secondary).

Citation

M. BOUDAOUD Abdelmadjid, BELLAOUAR DJAMEL, , (2020), "Representation of Integers: A nonclassical point of view", [national] Journal of logic and analysis , Nigel J. Cutland, Department of Mathematics, The University of York.

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2019-12-29

Nonclassical Study on certain Diophantine Inequalities involving Multiplicative Arithmetic Functions

ABSTRACT
This paper, for the most part, is in the framework of Internal Set Theory
(IST), where any real number must be infinitesimal, appreciable or unlim
ited; theses numbers are called standard or nonstandard. In particular,
any positive integer must be standard (limited) or nonstandard (unlim
ited). In the first part, we estimate for an unlimited positive integer n and
to an infinitesimal near, the values of some arithmetic functions of the
form f (n)
g (n) , where f and g are constructed using multiplicative functions.
Further, in the classical mathematics, several Diophantine inequalities
involving certain multiplicative arithmetic functions are studied.
Keywords: Diophantine Inequalities, Multiplicative Functions, Prime
Numbers, Internal Set Theory.

Citation

M. BOUDAOUD Abdelmadjid, Bellaouar Djamel, Boudaoud Said, , (2019-12-29), "Nonclassical Study on certain Diophantine Inequalities involving Multiplicative Arithmetic Functions", [national] Malaysian Journal of Mathematical Sciences , Universiti Putra Malaysia.

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2019-09-16

On a sequence formed by iterating a divisor operator

Let ℕ be the set of positive integers and let s ∈ ℕ. We denote by ds the arithmetic function given by ds(n) = (d(n))s, where d(n) is the number of positive divisors of n. Moreover, for every l, m ∈ ℕ we denote by δs,l,m (n) the sequence

We present classical and nonclassical notes on the sequence (δs, l, m(n))m⩾1, where l, n, s are understood as parameters.

Citation

M. BOUDAOUD Abdelmadjid, Bellaouar Djamel, Özen Özer (Turkey), , (2019-09-16), "On a sequence formed by iterating a divisor operator", [national] Czechoslovak Mathematical Journal , Springer

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2019

Representation of positive integers as a sum of distinct Tribonacci numbers

Let (Tm)m1 be the tribonacci sequence . We show that every integer N 1 can be written
as a sum of the terms mTm, where m runs over the set of strictly positive integers and m (m 1) are
either 1 or 0. The previous representation of N is unique if each time that we have m = 1 then at least
the two coecients directly following m are zero, i.e., m+1 = m+2 = 0.

Citation

M. BOUDAOUD Abdelmadjid, Badidja Salim, , (2019), "Representation of positive integers as a sum of distinct Tribonacci numbers", [international] 6th Inernational Arabe Conference On Mathematics And Computations , Jordan

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2018

The External Differential Equation

Our communication is placed in the enriched framework of nonstandard analysis. In the first place, we give definitions and properties of some notions such that : external set[1], neutrice, external number, ... .
In the second part, we review the work of some mathematicians concerning the introduction of several classical mathematical disciplines by using the above notions. In this sense we find, for example: External integration, External Differential Equation

Citation

M. BOUDAOUD Abdelmadjid, (2018), "The External Differential Equation", [national] TAMTAM , Université de Tlemcen

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Nonclassical study on certain diophantine equations

Nonclassical study on certain diophantine equations

Citation

M. BOUDAOUD Abdelmadjid, said.boudaoud@univ-msila.dz, , (2018), "Nonclassical study on certain diophantine equations", [national] WPAM'18, M'sila, dz , M'sila

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External numbers and their applications.

External numbers and their applications

Citation

M. BOUDAOUD Abdelmadjid, Imad Eddine Berrabah, , (2018), "External numbers and their applications.", [national] WPAM December 17-18, 2018 University of Msila , Université Mohamed Boudiaf de M'sila, Algérie.

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Simultaneous Factorization of integers Close to those of a system of Infinitely Large Integers

Let m 1 be an integer, and let (N1,N2, ...,Nm) be m?tuples of
unlimited positive integers. In this paper we are interested in the
factorization of the integers of the system (N1 ? r1,N2 ? r2, ...,Nm ? rm),
where r1, r2, ..., rm are relatively small integers. This objective is answered
in Theorems 4 and 5.

Citation

M. BOUDAOUD Abdelmadjid, (2018), "Simultaneous Factorization of integers Close to those of a system of Infinitely Large Integers", [international] ICOM, 3-6 July, 2018, Istanbul, Turkey. , Istanbul, Turkey

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Simultaneous Diophantine Approximation with Improvement due to the Farey Series

In this work we proof the following theorem which is, in addition to some
other lemmas, our main results :
Theorem. Let X = f(x1, t1) , (x2, t2) , ..., (xn, tn)g be a nite part of R R+,
then there exists 0 < #0 such that if 0 < # #0, then there exist rational numbers
pi q i=1,2,...,n satisfying 8< : xi ? pi q
#ti
#q ti ,i = 1, 2, ..., n.
It is clear that the condition #q ti for i = 1, 2, ..., n is equivalent to
#q t = Min
i=1,2,...,n
(ti ).

Citation

M. BOUDAOUD Abdelmadjid, (2018), "Simultaneous Diophantine Approximation with Improvement due to the Farey Series", [international] ICOM, 3-6 July, 2018, Istanbul, Turkey , Istanbul, Turkey

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2017-02-20

Unique representation of positive integers as a sum of distinct tribonacci numbers.

Abstract: Let (Tm)m≥1 be the tribonacci sequence. We show that every
integer N ≥ 1 can be written as a sum of the terms αm Tm, where m runs
over the set of strictly positive integers and αm (m ≥ 1) are either 1 or 0.
The previous representation of N is unique if each time that we have αm
= 1 then at least the two coefficients directly following αm are zero, i.e.,
αm+1 = αm+2 = 0.

Citation

M. BOUDAOUD Abdelmadjid, badidja Salim, , (2017-02-20), "Unique representation of positive integers as a sum of distinct tribonacci numbers.", [national] Journal of Mathematics and Statistics , Science publications

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