scholarly journals On the divisibility problem for one-relator monoids

1994 ◽  
Vol 55 (1) ◽  
pp. 3-7 ◽  
Author(s):  
S. I. Adian
Keyword(s):  
Divisibility Problem

A divisibility test for amateur discoverers

1970 ◽  
Vol 17 (1) ◽  
pp. 39-41
Author(s):  
Lewis Berenson
Keyword(s):  
Learning Opportunities ◽  
Average Student ◽  
Underlying Principle ◽  
Golden Opportunity ◽  
General Test ◽  
The Many ◽  
Divisibility Problem

The following is a description of an elementary divisibility test appropriate for students in grades 7 and 8 as an enrichment unit. It offers the average student a golden opportunity to join the ranks of the mathematical discoverers. The brighter student will enjoy refining the process and exploring the many open-ended learning opportunities that the test provides. The simplicity of this very general test as well as the ease with which its underlying principle can be grasped further enhance its attractiveness. The use of divisibility by nine and eleven in checks of computations is an added reason for introducing students to a test of divisibility. As illustrations, we shall consider divisibility by six and by eleven. The procedure may be applied in like manner to any divisibility problem.

A Divisibility Problem

1977 ◽  
Vol 56 (3) ◽  
pp. 291-294 ◽  
Author(s):  
Kenneth Lebensold
Keyword(s):  
Divisibility Problem

The Quotient Problem for Entire Functions

2018 ◽  
Vol 62 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Ji Guo
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Linear Recurrence ◽  
Natural Numbers ◽  
Complex Situation ◽  
Index Set ◽  
Diophantine Problem ◽  
Divisibility Problem ◽  
Recurrence Sequences

AbstractLet $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

A Divisibility Problem: 11082

2006 ◽  
Vol 113 (4) ◽  
pp. 370
Author(s):  
Wu Wei Chao
Keyword(s):  
Divisibility Problem

scholarly journals Divisibility of Selmer groups and class groups

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Kalyan Banerjee ◽  
Kalyan Chakraborty ◽  
Azizul Hoque
Keyword(s):  
Algebraic Geometry ◽  
Abelian Variety ◽  
Number Fields ◽  
The Other ◽  
Selmer Groups ◽  
Selmer Group ◽  
Quadratic Number ◽  
Class Groups ◽  
International Audience ◽  
Divisibility Problem

International audience In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.

scholarly journals Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

2021 ◽  
pp. 2150075
Author(s):  
Oktay K. Pashaev
Keyword(s):  
Analytic Functions ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Holomorphic Functions ◽  
Quantum Calculus ◽  
Infinite Hierarchy ◽  
Translation Operators ◽  
Number Formula ◽  
Divisibility Problem

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

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