Peculiarities of smoothly undulating number

his notes presents results related to divisibility or multiplicity between two numbers in the class of integers called  smoothly undulating numbers of the type uz[n]. The main result is to characterize and display types of divisors of some types of numbers uz[n], and we show an algorithm to determine the greatest common divisor between two numbers uz[n]

Reverse Divisors and Magic Numbers

The study examines the relationship between Ball's magic numbers and reverses divisors. These numbers are the source of beautiful and curious properties. Activities related to numbers can be a fun way to motivate mathematics students, while also enabling surprising analysis and connections.Comment: 19 page

A note on repunit number sequence

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit  rn by a prime p and its powers

On One-Zero numbers: a new Horadam-type sequence

In this paper, we present a new sequence of Horadam-type, which we call the One-Zero sequence. We study the recurrence equation and show the Binet formula. The aim of this study is to examine the properties of the aforementioned sequence. To this end, we have analyzed several classical identities, including the Tagiuri-Vajda and the Gelin-Cesàro identities. Additionally, we determine the partial sum of the terms of the One-Zero sequence

Tricomplex Ring with Complex coefficients

This study aims to explore and develop results related to the fundamental law of arithmetic within the framework of a commutative ring with unity. Specifically, it focuses on extending complex numbers to a vector space characterized by three complex coordinates, bridging foundational theoretical concepts with practical applications.Considering the extension of integer number sequences into other numerical sets, this research investigates a novel set of numbers. The extension of real numbers to higher dimensions, such as quaternions and octonions, has gained significance in physics due to their natural representation of certain symmetries in physical systems. In this work, we illustrate how the properties of complex numbers can be systematically leveraged to derive both the foundational basis and the multiplication rules for these advanced numerical systems.O objetivo deste estudo é explorar e desenvolver resultados relacionados com a lei fundamental da aritmética no contexto de um anel comutativo com unidade. Em particular, centra-se na extensão dos números complexos a um espaço vetorial caracterizado por três coordenadas complexas, fazendo a ponte entre os conceitos teóricos fundamentais e as aplicações práticas.Considerando a extensão de sequências de números inteiros a outros conjuntos de números, esta investigação investiga um novo conjunto de números. A extensão dos números reais a dimensões superiores, como os quaterniões e octiões, ganhou importância na física devido à sua representação natural de certas simetrias em sistemas físicos. Neste artigo, ilustramos como as propriedades dos números complexos podem ser sistematicamente utilizadas para derivar tanto a base fundamental como as regras de multiplicação destes sistemas numéricos avançados

Álgebras associativas Lie nilpotentes de classe 4

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Sejam K um anel associativo, comutativo e unitário e (K) X a K-álgebra associativa livre num conjunto não-vazio X de geradores livres. Defina um comutador normado à esquerda [x1;x2; : : : ;xn] por [a;b] = ab−ba e [a;b;c] = [ [a;b];c ] . Para n ≥ 2, seja T(n) o ideal bilateral em K(X) gerado pelos comutadores [a1;a2; : : : ;an] (ai ∈ K(X)). A álgebra quociente K(X)=T(n+1) pode ser vista como a K-álgebra universal associativa Lie nilpotente de classe n gerada por X. É fácil ver que o ideal T(2) é gerado, como um ideal bilateral em K(X), pelos comutadores [x1;x2] (xi ∈ X). É bem conhecido que o ideal T(3) é gerado pelos polinômios [x1;x2;x3] e [x1;x2][x3;x4]+[x1;x3][x2;x4] (xi ∈ X). Um conjunto similar de geradores para T(4) é também conhecido. O resultado principal do presente trabalho é exibir um conjunto semelhante de geradores para T(5). Nós provaremos que o ideal T(5) é gerado, como um ideal bilateral em K(X), pelos seguintes polinômios: [x1;x2;x3;x4;x5]; [x1;x2;x3][x4;x5;x6]; [x1;x2;x3][x4;x5;x6;x7]; [x1;x2][x3;x4;x5;x6]+[x6;x2][x3;x4;x5;x1]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] ) [x5;x6;x7]; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5;x6 ] ; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5 ] [x6;x7]+ [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x6 ] [x5;x7]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] )( [x5;x6][x7;x8]+[x5;x7][x6;x8] ) ; com xi ∈ X para todo i. Nós também descreveremos algumas componentes multilineares de Z(X)=L3 e Z(X)=L4, sendo Ln o n-ésimo termo da série central inferior de Z(X) visto como um anel de Lie . ______________________________________________________________________________ ABSTRACTLet K be a unital associative and commutative ring and let K(X) be the free associative K-algebra on a non-empty set X of free generators. Define a left-normed commutator [x1;x2; : : : ;xn] by [a;b] = ab−ba and [a;b;c] = [ [a;b];c ] . For n ≥ 2, let T(n) be the two-sided ideal in K(X) generated by all commutators [a1;a2; : : : ;an] (ai ∈ K(X)). The quotient algebra K(X)=T(n+1) can be viewed as the universal Lie nilpotent associative K-algebra of class n generated by X. It can be easily seen that the ideal T(2) is generated, as a two-sided ideal in K(X), by the commutators [x1;x2] (xi ∈ X). It is well-known that T(3) is generated by the polynomials [x1;x2;x3] and [x1;x2][x3;x4]+[x1;x3][x2;x4] (xi ∈ X). A similar generating set for T(4) is also known. The aim of the present work is to exhibit a similar generating set for T(5). We prove that the ideal T(5) is generated, as a two-sided ideal in K(X), by the following polynomials: [x1;x2;x3;x4;x5]; [x1;x2;x3][x4;x5;x6]; [x1;x2;x3][x4;x5;x6;x7]; [x1;x2][x3;x4;x5;x6]+[x6;x2][x3;x4;x5;x1]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] ) [x5;x6;x7]; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5;x6 ] ; [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x5 ] [x6;x7]+ [ [x1;x2][x3;x4]+[x1;x3][x2;x4];x6 ] [x5;x7]; ( [x1;x2][x3;x4]+[x1;x3][x2;x4] )( [x5;x6][x7;x8]+[x5;x7][x6;x8] ) ; where xi ∈ X for all i. We also describe some multilinear components of Z(X)=L3 and Z(X)=L4 where Ln is the n-th term of the lower central series of Z(X) viewed as a Lie ring