Concepts Of Number Theory Research Articles

Sistem Kriptografi Klasik Dengan Memanfaatkan Orde Dari Grup Titik Pada Kurva Eliptik Bentuk Montgomery

Elliptic curve cryptography is one of the application fields of algebra and number theory concepts. One form of elliptic curve cryptography is Montgomery elliptic curve cryptography. In this paper, a method for a classical cryptographic system be formulated, consisting of encryption and decryption involving twenty-six alphabetical letters which are mapped to points on an elliptic curve by utilizing the order of the point group on the Montgomery elliptic curve. Several examples of implementation in simple cases are given to verify the results.

On The Foundations of Fuzzy Number Theory and Fuzzy Diophantine Equations

Despite the great and rapid progress in the study of Fuzzy Logic and its applications in various scientific fields, it has not yet been used to build a consistent number theory like classical number theory. This research provides for the first time a conception of the concepts of number theory based on fuzzy logic and fuzzy membership functions, where it defines the division process, the fuzzy congruence, the greatest common divisor between integers with a fuzzy membership function. On the other hand, it presents many famous Diophantine equations formulated using fuzzy sets, in addition to many properties of fuzzy number theoretical systems, through many related theorems and accompanying illustrative examples. Also, in this research, we are raising many open research questions related to fuzzy number theory, which we believe will represent the future of progress in the study of this new mathematical branch.

Math Can Be Visual—Teaching and Understanding Arithmetical Functions through Visualization

Number theory is an area of mathematics not unknown to students majoring in mathematics teaching. As early as in junior high, they may encounter basic number theory concepts such as prime numbers, multiples, divisors, or even the fundamental theorem of arithmetic—the proof of which, however, will only be included in the university curriculum of mathematics teacher students. When teaching number theory, in addition to inspiring and motivating students, it is important to have them ask the right questions. Proofs are important, but so is the proper formulation of mathematical conjectures, not to mention intuitions regarding mathematical properties and relationships. These processes are facilitated by the software used, developed to visualize functions related to number theory. Our study aims to assess the relevance of visualization in the teaching and understanding of certain arithmetic functions among mathematics teacher students. Furthermore, we aim to investigate the practical usability of our visualization software, relying on student feedback. According to our results, students found the software useful, as it helped them to discover previously unknown relationships in the examined arithmetic functions. We may conclude our software fills a niche, with student feedback indicating its usefulness.

A generalization of multiple zeta values. Part 1: Recurrent sums

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

Strongly regular relations of arithmetic functions

A simple, but very useful concept in number theory is that of an arithmetic function. On the other hand, hyperstructure theory, first introduced by Marty, is a generalization of group theory and it is connected to different fields of mathematics. In our paper, we present a link between the theories of algebraic structures, hyperstructures and arithmetic functions. More precisely, we present hyperstructures of arithmetic functions and determine their fundamental groups and fundamental rings. Moreover, we study their complete parts and strongly regular relations.

An Algorithm to Find Square Root of Quadratic Residues over Finite Fields using Primitive Elements

Quadratic residue is an important concept in number theory because it has both theoretical and practical application in mathematics and other areas such as computer science and communication. We also have a same concept of quadratic residue in general finite fields. Finding a square root of a quadratic residue in finite fields is an essential problem in computational algebra. In this paper we present an algorithm of computing square root of quadratic residue in finite fields using a primitive element.

Sign domination in arithmetic graphs

Investigation of the graph, whose vertices are labelled with the natural numbers and adjacency among them influenced by concepts of number theory is of prime importance. This marks the collaboration of higher arithmetic with graph theory. In this paper, we compute the sign domination number of one such graph called Arithmetic graph. Also, we characterize the sign domination number with domination and efficient domination number of arithmetic graph.

Mystery Fractions

This activity engages students in problem solving while exploring key concepts of number theory, such as divisibility and divisibility tests, place value, fractions, and scale factors.

THE GRAPH OF DIVISOR FUNCTION $D(n)$

Starting from 1980, the concepts of Number Theory were used in Graph theory and several types of graphs have been introduced. In this sequel we have introduced divisor function graph and studied its properties. Further, divisor function subgraph, complete divisor function graph and Eulerian prop- erties have been discussed.

Some Domination Parameters of Arithmetic Vn Graph

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Nathanson [1] was the pioneer in introducing the concepts of Number Theory, particularly, the „Theory of Congruences‟ in Graph Theory, and paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Inspired by the interplay between Number Theory and Graph Theory several researchers in recent times are carrying out extensive studies on various Arithmetic graphs in which adjacency between vertices is defined through various arithmetic functions. Cayley Graphs are another class of graphs associated with elements of a group. If this group is associated with some Arithmetic function then the Cayley graph becomes an Arithmetic graph and in this paper we study the Efficient domination, Clique domination parameters of Arithmetic Vn Graphs.

Dynamic learning approach to studying curve stitching on a unit circle

In this article, we report a study on how students learn elementary number theory concepts via pre-constructed activities of curve stitching on a unit circle. By using a dynamic learning approach, students were able to observe and guess the patterns of curve- stitching figures and explore how they were related to the total number of points on the circle and the interval between any pair of connected points. The subjects were pre-service and in-service primary mathematics teachers majored in mathematics or information technology in a BEd programme offered by the Hong Kong Institute of Education. We also discuss some teaching and learning issues related to this topic.

Decoding a proof of Fermat's Little Theorem via script writing

Twelve participants were asked to decode – that is, interpret and make sense of – a given proof of Fermat's Little Theorem, and present it in a form of a script for a dialog between two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as ‘problematic’ in the proof and on how these issues were addressed. Affordances and limitations of this dialogic method of presenting proofs are exposed, by means of analyzing how the students’ correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogs. The difficulties identified by the participants are discussed in relation to past research on undergraduate students’ difficulties in proving and in understanding number theory concepts.

Basic Edge Dominating Functions of Quadratic Residue Cayley Graphs

Graph Theory has been realized as one of the most useful branches of Mathematics of recent origin, finding widest applications in all most all branches of sciences, social sciences, and engineering and computer science. Nathanson[8] was the pioneer in introducing the concepts of NumberTheory, particularly, the “Theory of congruences” i n Graph Theory, thus paving way for the emergence of a new class of graphs, namely, “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an arithmetic graph. Quadratic residue is an arithmetic function which is defined by: Let p be an odd prime and n, a positive integer such that n 0 (mod p). If the quadratic congruence, ) (mod 2 p n x  has a solution then, n is called a quadratic residue mod p. The Quadratic Residue Cayley graph G(Zp , Q), is the Cayley graph associated with the quadratic residue function. The theory of basic edge dominating functions in Quadratic Residue Cayley Graphs is useful in the selection of modes, those are require to focus on the development of some connected systems like market management, operating system, banking , infrastructure system etc.

Some Domination Parameters of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Nathanson [1] paved the way for the emergence of a new class of graphs, namely Arithmetic Graphs by introducing the concepts of Number Theory, particularly, the Theory of Congruences in Graph Theory. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph. Inspired by the interplay between Number Theory and Graph Theory several researchers in recent times are carrying out extensive studies on various Arithmetic graphs in which adjacency between vertices is defined through various arithmetic functions.

Some Properties of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paving way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph. In this paper, we present some results related to basic properties of direct product graphs of Euler totient Cayley graphs with Arithmetic graph.

Some Domination Parameters of Arithmetic Graph Vn

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Nathanson [1] was the pioneer in introducing the concepts of Number Theory, particularly, the „Theory of Congruences‟ in Graph Theory, thus paving way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Inspired by the interplay between Number Theory and Graph Theory several researchers in recent times are carrying out extensive studies on various Arithmetic graphs in which adjacency between vertices is defined through various arithmetic functions.

Modelling problem-solving situations into number theory tasks: The route towards generalisation

This paper examines the way two 10th graders cope with a non-standard generalisation problem that involves elementary concepts of number theory (more specifically linear Diophantine equations) in the geometrical context of a rectangle’s area. Emphasis is given on how the students’ past experience of problem solving (expressed through interplay among different modes of thinking and actions that show executive control and decision-making skills) supported them in their route towards generalisation.

Building Numbers from Primes

Use building blocks to create a visual model for prime factorizations. Students can explore many concepts of number theory, including the relationship between greatest common factors and least common multiples.

Explorations with 142857: Connecting the Elementary with the Advanced

Explorations with the cyclic number for students and teachers help connect the mathematics taught in school with algebra and number theory concepts learned in college. Additional ways to manipulate sequential numbers and how they relate create new sequences as well as graphical and visual representations are given.

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.