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Vector Partitions, Jennifer French 2018 East Tennessee State University

Vector Partitions, Jennifer French

Electronic Theses and Dissertations

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The ...


The Distribution Of Totally Positive Integers In Totally Real Number Fields, Tianyi Mao 2018 The Graduate Center, City University of New York

The Distribution Of Totally Positive Integers In Totally Real Number Fields, Tianyi Mao

All Dissertations, Theses, and Capstone Projects

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the number field is quadratic, Beck also proved a mean value result using the continued fraction expansions of quadratic irrationals. We generalize Beck’s result to higher moments. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the ...


Quantum Attacks On Modern Cryptography And Post-Quantum Cryptosystems, Zachary Marron 2018 Liberty University

Quantum Attacks On Modern Cryptography And Post-Quantum Cryptosystems, Zachary Marron

Senior Honors Theses

Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial ...


Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, Sarah Riccio 2018 Sacred Heart University

Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, Sarah Riccio

Mathematics Undergraduate Publications

In this paper, three topics in number theory will be explored: Niven Numbers, the Factorial Triangle, and Erdos's Conjecture . For each of these topics, the goal is for us to find patterns within the numbers which help us determine all possible values in each category. We will look at two digit Niven Numbers and the set that they belong to, the alternating summation of the rows of the Factorial Triangle, and the unit fractions whose sum is the basis of Erdos' Conjecture.


Monomial Progenitors And Related Topics, Madai Obaid Alnominy 2018 California State University - San Bernardino

Monomial Progenitors And Related Topics, Madai Obaid Alnominy

Electronic Theses, Projects, and Dissertations

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 2 ...


The Rsa Cryptosystem, Rodrigo Iglesias 2018 The University of Akron

The Rsa Cryptosystem, Rodrigo Iglesias

Honors Research Projects

This paper intends to present an overview of the RSA cryptosystem. Cryptosystems are mathematical algorithms that disguise information so that only the people for whom the information is intended can read it. The invention of the RSA cryptosystem in 1977 was a significant event in the history of cryptosystems. We will describe in detail how the RSA cryptosystem works and then illustrate the process with a realistic example using fictional characters. In addition, we will discuss how cryptosystems worked prior to the invention of RSA and the advantage of using RSA over any of the previous cryptosystems. This will help ...


An Algorithm To Determine All Odd Primitive Abundant Numbers With D Divisors, Jacob Liddy 2018 The University of Akron

An Algorithm To Determine All Odd Primitive Abundant Numbers With D Divisors, Jacob Liddy

Honors Research Projects

An abundant number is said to be primitive if none of its proper divisors are abundant. Dickson proved that for an arbitrary positive integer d there exists only finitely many odd primitive abundant numbers having exactly d prime divisors. In this paper we describe a fast algorithm that finds all primitive odd numbers with d unique prime divisors. We use this algorithm to find all the number of odd primitive abundant numbers with 6 unique Divisors. We use this algorithm to prove that an odd weird number must have at least 6 prime divisors.


Theory Of Algebraic Numbers, Foreword & Chapters 1-4, By Kurt Hensel (1908), Kurt Hensel, Daniel E. Otero 2018 Xavier University - Cincinnati

Theory Of Algebraic Numbers, Foreword & Chapters 1-4, By Kurt Hensel (1908), Kurt Hensel, Daniel E. Otero

2018, March 23-24 ORESME Reading Group Meeting

English translation of the first four chapters of Kurt Hensel's Theorie der Algebraischen Zahlen (1908), by Daniel E. Otero.


On The Density Of The Odd Values Of The Partition Function, Samuel Judge 2018 Michigan Technological University

On The Density Of The Odd Values Of The Partition Function, Samuel Judge

Dissertations, Master's Theses and Master's Reports

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities ...


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

Generating Pythagorean Triples: A Gnomonic Exploration, Janet Heine Barnett

Number Theory

No abstract provided.


The Pell Equation In India, Toke Knudson, Keith Jones 2017 Ursinus College

The Pell Equation In India, Toke Knudson, Keith Jones

Number Theory

No abstract provided.


Asymptotic Counting Formulas For Markoff-Hurwitz Tuples, Ryan Ronan 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, Karl Levy 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, Ricardo Conceição, R. Gondim, M. Rodriguez 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, David Skidmore 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 20.25log2(α)3 + 16.5log2(α)2 + 6log ...


Transfinite Ordinal Arithmetic, James Roger Clark 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, Sara R. Katz 2017 Essex Street Academy

A Math Poem, Sara R. Katz

Journal of Humanistic Mathematics

No abstract provided.


Primes, Divisibility, And Factoring, Dominic Klyve 2017 Central Washington University

Primes, Divisibility, And Factoring, Dominic Klyve

Number Theory

No abstract provided.


Construction Of The Figurate Numbers, Jerry Lodder 2017 New Mexico State University

Construction Of The Figurate Numbers, Jerry Lodder

Number Theory

No abstract provided.


Babylonian Numeration, Dominic Klyve 2017 Central Washington University

Babylonian Numeration, Dominic Klyve

Number Theory

No abstract provided.


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