# Number Theory Commons™

## All Articles in Number Theory

429 full-text articles. Page 2 of 17.

Applying The Data: Predictive Analytics In Sport, 2020 University of Washington, Tacoma

#### Applying The Data: Predictive Analytics In Sport, Anthony Teeter, Margo Bergman

##### Access*: Interdisciplinary Journal of Student Research and Scholarship

The history of wagering predictions and their impact on wide reaching disciplines such as statistics and economics dates to at least the 1700’s, if not before. Predicting the outcomes of sports is a multibillion-dollar business that capitalizes on these tools but is in constant development with the addition of big data analytics methods. Sportsline.com, a popular website for fantasy sports leagues, provides odds predictions in multiple sports, produces proprietary computer models of both winning and losing teams, and provides specific point estimates. To test likely candidates for inclusion in these prediction algorithms, the authors developed a computer model ...

Decision Making On Teachers’ Adaptation To Cybergogy In Saturated Interval- Valued Refined Neutrosophic Overset /Underset /Offset Environment, 2020 University of New Mexico

#### Decision Making On Teachers’ Adaptation To Cybergogy In Saturated Interval- Valued Refined Neutrosophic Overset /Underset /Offset Environment, Florentin Smarandache, Nivetha Martin, Priya R.

##### Mathematics and Statistics Faculty and Staff Publications

Neutrosophic overset, neutrosophic underset and neutrosophic offset introduced by Smarandache are the special kinds of neutrosophic sets with values beyond the range [0,1] and these sets are pragmatic in nature as it represents the real life situations. This paper introduces the concept of saturated refined neutrosophic sets and extends the same to the special kinds of neutrosophic sets. The proposed concept is applied in decision making on Teacher’s adaptation to cybergogy. The decision making environment is characterized by different types of teachers, online teaching skills and various training methods. Fuzzy relation is used to match the most suitable ...

Arithmetical Structures On Paths With A Doubled Edge, 2020 Gettysburg College

#### Arithmetical Structures On Paths With A Doubled Edge, Darren B. Glass, Joshua R. Wagner

##### Math Faculty Publications

An arithmetical structure on a graph is given by a labeling of the vertices that satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical structures for graphs in these families.

Small Gaps Between Almost Primes, The Parity Problem, And Some Conjectures Of Erdős On Consecutive Integers Ii, 2020 San Jose State University

#### Small Gaps Between Almost Primes, The Parity Problem, And Some Conjectures Of Erdős On Consecutive Integers Ii, Daniel A. Goldston, Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, Cem Y. Yıldırım

##### Faculty Publications

We show that for any positive integer n, there is some fixed A such that d(x) = d(x +n) = A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x +n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x > 1is the multiset of nonzero exponents which appear in the prime factorization of x.

2020 University of St. Thomas - Houston

#### Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, Mary K. Flagg

##### Number Theory

No abstract provided.

Harmony Amid Chaos, 2020 Olivet Nazarene University

#### Harmony Amid Chaos, Drew Schaffner

##### Pence-Boyce STEM Student Scholarship

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as ...

Multiparty Non-Interactive Key Exchange And More From Isogenies On Elliptic Curves, 2020 Stanford University

#### Multiparty Non-Interactive Key Exchange And More From Isogenies On Elliptic Curves, Dan Boneh, Darren B. Glass, Daniel Krashen, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, Mark Zhandry

##### Math Faculty Publications

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety.

Our framework builds a cryptographic invariant ...

Quadratic Packing Polynomials On Sectors Of R2, 2020 The Graduate Center, City University of New York

#### Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek

##### Dissertations, Theses, and Capstone Projects

A result by Fueter-Pólya states that the only quadratic polynomials that bijectively map the integral lattice points of the first quadrant onto the non-negative integers are the two Cantor polynomials. We study the more general case of bijective mappings of quadratic polynomials from the lattice points of sectors defined as the convex hull of two rays emanating from the origin, one of which falls along the x-axis, the other being defined by some vector. The sector is considered rational or irrational according to whether this vector can be written with rational coordinates or not. We show that the existence of ...

2020 The Graduate Center, City University of New York

#### Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan

##### Dissertations, Theses, and Capstone Projects

Harris and Kudla (2004) proved a conjecture of Jacquet, that the central value of a triple product L-function does not vanish if and only if there exists a quaternion algebra over which a period integral of three corresponding automorphic forms does not vanish. Moreover, Gross and Kudla (1992) established an explicit identity relating central L-values and period integrals (which are finite sums in their case), when the cusp forms are of prime levels and weight 2. Böcherer, Schulze-Pillot (1996) and Watson (2002) generalized this identity to more general levels and weights, and Ichino (2008) proved an adelic period formula which ...

The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, 2020 CUNY Bernard M Baruch College

#### The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes

##### Student Theses

For a pair of quadratic integers n and m chosen randomly, uniformly, and independently from the set of quadratic integers of norm x or less, we calculate the probability that the greatest common divisor of (n,m) is k. We also calculate the expected norm of the greatest common divisor (n,m) as x tends to infinity, with explicit error terms. We determine the probability and expected norm of the greatest common divisor for quadratic integer rings that are unique factorization domains. We also outline a method to determine the probability and expected norm of the greatest common divisor of ...

An In-Depth Look At P-Adic Numbers, 2020 CUNY New York City College of Technology

#### An In-Depth Look At P-Adic Numbers, Xiaona Zhou

##### Publications and Research

In this study, we consider $p$-adic numbers. We will also study the $p$-adic norm representation of real number, which is defined as $\mathbb{Q}_p = \{\sum_{j=m}^{\infty }a_j p^j: a_j \in \mathbb{D}_p, m\in\mathbb{Z}, a_m\neq 0\} \cup \{0\}$, where $p$ is a prime number. We explore properties of the $p$-adics by using examples. In particular, we will show that $\sqrt{6},i \in \mathbb{Q}_5$ and $\sqrt{2} \in \mathbb{Q}_7$. $p$-adic numbers have a wide range of applicationsnin fields such as string theory, quantum mechanics ...

Calculating Infinite Series Using Parseval's Identity, 2020 University of Maine

#### Calculating Infinite Series Using Parseval's Identity, James R. Poulin

##### Electronic Theses and Dissertations

Parseval's identity is an equality from Fourier analysis that relates an infinite series over the integers to an integral over an interval, which can be used to evaluate the exact value of some classes of infinite series. We compute the exact value of the Riemann zeta function at the positive even integers using the identity, and then we use it to compute the exact value of an infinite series whose summand is a rational function summable over the integers.

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, 2020 Louisiana State University and Agricultural and Mechanical College

#### Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

##### LSU Doctoral Dissertations

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit ...

Local-Global Principles For Diophantine Equations, 2020 Liberty University

#### Local-Global Principles For Diophantine Equations, Benjamin Barham

##### Senior Honors Theses

The real number field, denoted , is the most well-known extension field of ℚ, the field of rational numbers, but it is not the only one. For each prime p, there exists an extension field ℚp of ℚ, and these fields, known as the p-adic fields, have some properties substantially different from . In this paper, we construct the p-adic numbers from the ground up and discuss the local-global principle, which concerns connections between solutions of equations found in ℚ and in ℚp. We state the Hasse-Minkowski theorem, which addresses a type of Diophantine equation to ...

Symmetric Presentations And Related Topics, 2020 California State University, San Bernardino

#### Symmetric Presentations And Related Topics, Mayra Mcgrath

##### Electronic Theses, Projects, and Dissertations

In this thesis, we have investigated several permutation and monomialprogenitors for finite images. We have found original symmetric presen-tations for several important non-abelian simple groups, including lineargroups, unitary groups, alternating groups, and sporadic simple groups.We have found a number of finite images, including : L(2,41), PSL(2,11)×2, L(2,8), and L(2,19), as homomorphic images of the permutation progenitors. We have also found PGL(2,16) : 2 =Aut(PSL(2,16)) and PSL(2,16) as homomorphic images of monomial progenitors. We have performed manual double coset enumeration of finte images. In addition, we ...

2020 California State University, San Bernardino

#### Collaboration (Reacting To The Past/Math/History/Writing), James Hayashi

##### Q2S Enhancing Pedagogy

This is an assignment for a Freshman level course in the College of Natural Science. By the end students will have an understanding of valid research, collaboration and communication skills. Faculty that chooses to use this assignment will be preparing students for an active learning environment, and understanding a “Big Idea”, valid research, technology and communication skills.

Faculty should give an example of what is valid research. As students are completing this assignment mini deadlines (check-ins) shall be set. With the check-ins for this assignment focus on how the group will communicate the check point and the collaboration.

The focus ...

Arithmetic Of Binary Cubic Forms, 2020 The Graduate Center, City University of New York

#### Arithmetic Of Binary Cubic Forms, Gennady Yassiyevich

##### Dissertations, Theses, and Capstone Projects

The goal of the thesis is to establish composition laws for binary cubic forms. We will describe both the rational law and the integral law. The rational law of composition is easier to describe. Under certain conditions, which will be stated in the thesis, the integral law of composition will follow from the rational law. The end result is a new way of looking at the law of composition for integral binary cubic forms.

On Consecutive Triples Of Powerful Numbers, 2020 Oliver Wendell Holmes Junior High School, Davis, CA

#### On Consecutive Triples Of Powerful Numbers, Edward Beckon

No abstract provided.

Consecutive Prime And Highly Total Prime Labeling In Graphs, 2020 La Salle University