Generating Pythagorean Triples: A Gnomonic Exploration, 2017 Colorado State University-Pueblo

#### Generating Pythagorean Triples: A Gnomonic Exploration, Janet Heine Barnett

*Number Theory*

No abstract provided.

The Pell Equation In India, 2017 Ursinus College

#### The Pell Equation In India, Toke Knudson, Keith Jones

*Number Theory*

No abstract provided.

Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, 2017 The Graduate Center, City University of New York

#### Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan

*All Dissertations, Theses, and Capstone Projects*

The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff's celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of ...

Some Results In Combinatorial Number Theory, 2017 The Graduate Center, City University of New York

#### Some Results In Combinatorial Number Theory, Karl Levy

*All Dissertations, Theses, and Capstone Projects*

The first chapter establishes results concerning equidistributed sequences of numbers. For a given $d\in\mathbb{N}$, $s(d)$ is the largest $N\in\mathbb{N}$ for which there is an $N$-regular sequence with $d$ irregularities. We compute lower bounds for $s(d)$ for $d\leq 10000$ and then demonstrate lower and upper bounds $\left\lfloor\sqrt{4d+895}+1\right\rfloor\leq s(d)< 24801d^{3} + 942d^{2} + 3$ for all $d\geq 1$. In the second chapter we ask if $Q(x)\in\mathbb{R}[x]$ is a degree $d$ polynomial such that for $x\in[x_k]=\{x_1,\cdots,x_k\}$ we have $|Q(x)|\leq 1$, then how big can its lead coefficient be? We prove that there is a unique polynomial, which we call $L_{d,[x_k]}(x)$, with maximum lead coefficient under these constraints and construct an algorithm that generates $L_{d,[x_k]}(x)$.

On A Frobenius Problem For Polynomials, 2017 Gettysburg College

#### On A Frobenius Problem For Polynomials, Ricardo Conceição, R. Gondim, M. Rodriguez

*Math Faculty Publications*

We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~*k*. Similar to the classical problem we show that the *n* = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to *k*[*t*] and give an algorithm to solve the Frobenius problem for polynomials over a field *k* of sufficiently large size.

Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, 2017 Utah State University

#### Efficiently Representing The Integer Factorization Problem Using Binary Decision Diagrams, David Skidmore

*All Graduate Plan B and other Reports*

Let p be a prime positive integer and let α be a positive integer greater than 1. A method is given to reduce the problem of finding a nontrivial factorization of α to the problem of finding a solution to a system of modulo p polynomial congruences where each variable in the system is constrained to the set {0,...,p − 1}. In the case that p = 2 it is shown that each polynomial in the system can be represented by an ordered binary decision diagram with size less than 20.25log_{2}(α)^{3} + 16.5log_{2}(α)^{2} + 6log ...

Transfinite Ordinal Arithmetic, 2017 Governors State University

#### Transfinite Ordinal Arithmetic, James Roger Clark

*All Student Theses*

Following the literature from the origin of Set Theory in the late 19th century to more current times, an arithmetic of finite and transfinite ordinal numbers is outlined. The concept of a set is outlined and directed to the understanding that an ordinal, a special kind of number, is a particular kind of well-ordered set. From this, the idea of counting ordinals is introduced. With the fundamental notion of counting addressed: then addition, multiplication, and exponentiation are defined and developed by established fundamentals of Set Theory. Many known theorems are based upon this foundation. Ultimately, as part of the conclusion ...

A Math Poem, 2017 Essex Street Academy

Primes, Divisibility, And Factoring, 2017 Central Washington University

#### Primes, Divisibility, And Factoring, Dominic Klyve

*Number Theory*

No abstract provided.

Construction Of The Figurate Numbers, 2017 New Mexico State University

#### Construction Of The Figurate Numbers, Jerry Lodder

*Number Theory*

No abstract provided.

Babylonian Numeration, 2017 Central Washington University

Pascal's Triangle And Mathematical Induction, 2017 New Mexico State University

#### Pascal's Triangle And Mathematical Induction, Jerry Lodder

*Number Theory*

No abstract provided.

Gaussian Integers And Dedekind's Creation Of An Ideal: A Number Theory Project, 2017 Colorado State University-Pueblo

#### Gaussian Integers And Dedekind's Creation Of An Ideal: A Number Theory Project, Janet Heine Barnett

*Number Theory*

No abstract provided.

Diophantine Approximation And The Atypical Numbers Of Nathanson And O'Bryant, 2017 The Graduate Center, City University of New York

#### Diophantine Approximation And The Atypical Numbers Of Nathanson And O'Bryant, David Seff

*All Dissertations, Theses, and Capstone Projects*

For any positive real number $\theta > 1$, and any natural number $n$, it is obvious that sequence $\theta^{1/n}$ goes to 1. Nathanson and O'Bryant studied the details of this convergence and discovered some truly amazing properties. One critical discovery is that for almost all $n$, $\displaystyle\floor{\frac{1}{\fp{\theta^{1/n}}}}$ is equal to $\displaystyle\floor{\frac{n}{\log\theta}-\frac{1}{2}}$, the exceptions, when $n > \log_2 \theta$, being termed atypical $n$ (the set of which for fixed $\theta$ being named $\mcA_\theta$), and that for $\log\theta$ rational, the number of atypical $n ...

Counting Rational Points, Integral Points, Fields, And Hypersurfaces, 2017 The Graduate Center, City University of New York

#### Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther

*All Dissertations, Theses, and Capstone Projects*

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5 ...

Algorithmic Factorization Of Polynomials Over Number Fields, 2017 Rose-Hulman Institute of Technology

#### Algorithmic Factorization Of Polynomials Over Number Fields, Christian Schulz

*Mathematical Sciences Technical Reports (MSTR)*

The problem of exact polynomial factorization, in other words expressing a polynomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of ...

From Simplest Recursion To The Recursion Of Generalizations Of Cross Polytope Numbers, 2017 Kennesaw State University

#### From Simplest Recursion To The Recursion Of Generalizations Of Cross Polytope Numbers, Yutong Yang

*Honors College Capstones and Theses*

My research project involves investigations in the mathematical field of combinatorics. The research study will be based on the results of Professors Steven Edwards and William Griffiths, who recently found a new formula for the cross-polytope numbers. My topic will be focused on "Generalizations of cross-polytope numbers". It will include the proofs of the combinatorics results in Dr. Edwards and Dr. Griffiths' recently published paper. $E(n,m)$ and $O(n,m)$, the even terms and odd terms for Dr. Edward's original combinatorial expression, are two distinct combinatorial expressions that are in fact equal. But there is no obvious ...

Roman Domination In Complementary Prisms, 2017 East Tennessee State University

#### Roman Domination In Complementary Prisms, Alawi I. Alhashim

*Electronic Theses and Dissertations*

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number ...

On The Reality Of Mathematics, 2017 Southeastern University - Lakeland

#### On The Reality Of Mathematics, Brendan Ortmann

*Selected Student Publications*

Mathematics is an integral cornerstone of science and society at large, and its implications and derivations should be considered. That mathematics is frequently abstracted from reality is a notion not countered, but one must also think upon its physical basis as well. By segmenting mathematics into its different, abstract philosophies and real-world applications, this paper seeks to peer into the space that mathematics seems to fill; that is, to understand *how* and *why* it works. Under mathematical theory, Platonism, Nominalism, and Fictionalism are analyzed for their validity and their shortcomings, in addition to the evaluation of infinities and infinitesimals, to ...

On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, 2017 The University of Western Ontario

#### On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi

*Electronic Thesis and Dissertation Repository*

The objective of the study is to investigate the behaviour of the inner products of vector-valued Poincare series, for large weight, associated to submanifolds of a quotient of the complex unit ball and how vector-valued automorphic forms could be constructed via Poincare series. In addition, it provides a proof of that vector-valued Poincare series on an irreducible bounded symmetric domain span the space of vector-valued automorphic forms.