University of Nebraska - Lincoln
University of Connecticut - Storrs
Eastern Illinois University
Parkland College
Georgia Southern University
Ursinus College
Vector Partitions,
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,
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,
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,
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,
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,
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,
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),
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,
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,
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 20.25log2(α)3 + 16.5log2(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
