Number Theory Commons^™

The Fibonacci Sequence And Hosoya's Triangle, Jeffrey Lee Smith

Theses Digitization Project

The purpose of this thesis is to study the Fibonacci sequence in a context many are unfamiliar with. A triangular array of numbers, similar looking to Pascal's triangle, was constructed a few decades ago and is called Hosoya's triangle. Each element within the triangle is created using Fibonacci numbers.

Comparing The Algebraic And Analytical Properties Of P-Adic Numbers With Real Numbers, Joseph Colton Wilson

Theses Digitization Project

This study will provide a glimpse into the world of p-adic numbers, which encompasses a different way to measure the distance between rational numbers. Simple calculations and surprising results are examined to help familiarize the reader to the new space.

Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, Tracy Layden

Scripps Senior Theses

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Gaussian Amicable Pairs, Ranthony Ashley Clark

Online Theses and Dissertations

Amicable pairs are two integers where the sum of the proper divisors of one is the other and vice versa. Since the Gaussian integers have many of the properties of the regular integers, we sought to discover whether there exist any pairs of Gaussian integers with the same property. It turns out that they do exist. In fact, some of the normal amicable pairs carry over as Gaussian amicable pairs. Also discovered are pairs that have a complex part.

Set Theoretic Approach To Algebraic Structures In Mathematics - A Revelation, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out how sets in algebraic structure can be used to construct most generalized algebraic structures, like set linear algebra/vector space, set ideals in rings and semigroups. This sort of study is not only innovative but infact very helpful in cases instead of working with a large data we can work with a considerably small data. Thus instead of working with a vector space or a linear algebra V over a field F we can work with a subset in V and a needed subset in F, this can save both time and economy. The concept …

More On Intuitionistic Neutrosophic Soft Sets, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Intuitionistic Neutrosophic Soft Set theory proposed by S. Broumi and F. Samarandache [28], has been regarded as an effective mathematical tool to deal with uncertainties. In this paper new operations on intuitionistic neutrosophic soft sets have been introduced . Some results relating to the properties of these operations have been established. Moreover ,we illustrate their interconnections between each other.

Subset Non Associative Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of non associative topological space is new and innovative. In general topological spaces are defined as union and intersection of subsets of a set X. In this book authors for the first time define non associative topological spaces using subsets of groupoids or subsets of loops or subsets of groupoid rings or subsets of loop rings. This study leads to several interesting results in this direction.

Over hundred problems on non associative topological spaces using of subsets of loops or groupoids is suggested at the end of chapter two. Also conditions for these non associative subset topological spaces …

Subset Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of constructing non associative algebraic structures using subsets of a groupoid. Thus subset groupoids are constructed using groupoids or loops. Even if we use subsets of loops still the algebraic structure we get with it is only a groupoid. However we can get a proper subset of it to be a subset loop which will be isomorphic with the loop which was used in the construction of the subset groupoid. To the best of the authors’ knowledge this is the first time non associative algebraic structures are constructed using subsets. We get …

Variance On Topics Of Plane Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

This book contains 21 papers of plane geometry. It deals with various topics, such as: quasi-isogonal cevians, nedians, polar of a point with respect to a circle, anti-bisector, aalsonti-symmedian, anti-height and their isogonal. A nedian is a line segment that has its origin in a triangle’s vertex and divides the opposite side in Q equal segments. The papers also study distances between remarkable points in the 2D-geometry, the circumscribed octagon and the inscribable octagon, the circles adjointly ex-inscribed associated to a triangle, and several classical results such as: Carnot circles, Euler’s line, Desargues theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s theorem, …

Subset Polynomial Semirings And Subset Matrix Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of subset polynomial semirings and subset matrix semirings. The study of algebraic structures using subsets were recently carried out by the authors. Here we define the notion of subset row matrices, subset column matrices and subset m × n matrices. Study of this kind is developed in chapter two of this book. If we use subsets of a set X; say P(X), the power set of the set X....

Hence if P(X) is replaced by a group or a semigroup we get the subset matrix to be only a subset matrix semigroup. If …

Subset Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of groupoids is meager and we have recently introduced the new notion of subset groupoids and have studied them. It is interesting to keep on record that interval groupoids have been studied by us in 2010. Further when the subsets of a loop are taken they also form only a subset groupoid and not a subset loop. Thus we do not have the concept of subset interval loop they only form a subset interval groupoid. Special elements like subset interval zero divisors, subset interval idempotents and subset interval units are studied. Concept of subset interval groupoid homomorphism is …

Algebraic Structures Using Subsets, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a …

Algebraic Structures Using [0,N), Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n. We see [0, n) under both max and min operation is a semiring. [0, n) under + and × is not in general a ring. We define S = {[0, n), +, ×} to be a pseudo special ring as the distributive law is …

Intuitionistic Neutrosophic Soft Set, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we study the concept of intuitionistic neutrosophic set of Bhowmik and Pal. We have introduced this concept in soft sets and defined intuitionistic neutrosophic soft set. Some definitions and operations have been introduced on intuitionistic neutrosophic soft set. Some properties of this concept have been established.

A New Approach For Computing Wz Factorization, Effat Golpar-Raboky

Applications and Applied Mathematics: An International Journal (AAM)

Linear systems arise frequently in scientific and engineering computing. Various serial and parallel algorithms have been introduced for their solution. This paper seeks to compute the WZ and the ZW factorizations of a nonsingular matrix A using the right inverse of nested submatrices of A. We introduce two new matrix factorizations, the QZ and the QW factorizations, and compute the factorizations using our proposed approach.

Generalizations Of Two Statistics On Linear Tilings, Toufik Mansour, Mark Shattuck

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we study generalizations of two well-known statistics on linear square-and-domino tilings by considering only those dominos whose right half covers a multiple of 􀝇, where 􀝇 is a fixed positive integer. Using the method of generating functions, we derive explicit expressions for the joint distribution polynomials of the two statistics with the statistic that records the number of squares in a tiling. In this way, we obtain two families of q -generalizations of the Fibonacci polynomials. When 􀝇 􀵌 1, our formulas reduce to known results concerning previous statistics. Special attention is payed to the case 􀝇 …

A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Q -Analogs Of Identities Involving Harmonic Numbers And Binomial Coefficients, Toufik Mansour, Mark Shattuck, Chunwei Song

Applications and Applied Mathematics: An International Journal (AAM)

Recently, McCarthy presented two algebraic identities involving binomial coefficients and harmonic numbers, one of which generalizes an identity used to prove the Apéry number supercongruence. In 2008, Prodinger provided human proofs of identities initially obtained by Osburn and Schneider using the computer program Sigma. In this paper, we establish q -analogs of a fair number of the identities appearing in McCarthy (Integers 11 (2011): A37) and Prodinger (Integers 8 (2008): A10) by making use of q -partial fractions.

Elliptic Curves Of High Rank, Cecylia Bocovich

Mathematics, Statistics, and Computer Science Honors Projects

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y^₂ = x³ + ax + b, where a and b are coefficients that satisfy the property 4a³ + 27b² = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic …

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

All HMC Faculty Publications and Research

We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.