Sums Involving The Number Of Distinct Prime Factors Function, 2018 University of Maryland, College Park

#### Sums Involving The Number Of Distinct Prime Factors Function, Tanay Wakhare

*Rose-Hulman Undergraduate Mathematics Journal*

We find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria for these series. The approach of this paper is to use the theory of symmetric functions to derive identities for the elementary symmetric functions, then apply these identities to arbitrary primes and values of multiplicative functions evaluated at primes. This allows us to reinterpret sums over symmetric polynomials as divisor sums and sums over the natural numbers.

On Orders Of Elliptic Curves Over Finite Fields, 2018 Columbia University

#### On Orders Of Elliptic Curves Over Finite Fields, Yujin H. Kim

*Rose-Hulman Undergraduate Mathematics Journal*

In this work, we completely characterize by $j$-invariant the number of orders of elliptic curves over all finite fields $F_{p^r}$ using combinatorial arguments and elementary number theory. Whenever possible, we state and prove exactly which orders can be taken on.

Modern Cryptography, 2018 California State University - San Bernardino

#### Modern Cryptography, Samuel Lopez

*Electronic Theses, Projects, and Dissertations*

We live in an age where we willingly provide our social security number, credit card information, home address and countless other sensitive information over the Internet. Whether you are buying a phone case from Amazon, sending in an on-line job application, or logging into your on-line bank account, you trust that the sensitive data you enter is secure. As our technology and computing power become more sophisticated, so do the tools used by potential hackers to our information. In this paper, the underlying mathematics within ciphers will be looked at to understand the security of modern ciphers.

An extremely important ...

Secure Multiparty Protocol For Differentially-Private Data Release, 2018 Boise State University

#### Secure Multiparty Protocol For Differentially-Private Data Release, Anthony Harris

*Boise State University Theses and Dissertations*

In the era where big data is the new norm, a higher emphasis has been placed on models which guarantees the release and exchange of data. The need for privacy-preserving data arose as more sophisticated data-mining techniques led to breaches of sensitive information. In this thesis, we present a secure multiparty protocol for the purpose of integrating multiple datasets simultaneously such that the contents of each dataset is not revealed to any of the data owners, and the contents of the integrated data do not compromise individual’s privacy. We utilize privacy by simulation to prove that the protocol is ...

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 ...

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 ...

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 M_{11}, HS × D_{5}, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L_{2}(149) as homomorphic images of the monomial progenitors 11*^{4} :_{m} (5 :4), 5*^{6 } :_{m} S_{5} and 149*^{2 } :_{m } D_{37}. We have also discovered 2^{4} : S_{3} × C_{2}, 2 ...

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 ...

Equidimensional Adic Eigenvarieties For Groups With Discrete Series, 2018 Illinois Mathematics and Science Academy

#### Equidimensional Adic Eigenvarieties For Groups With Discrete Series, Daniel Robert Gulotta '03

*Doctoral Dissertations*

We extend Urban's construction of eigenvarieties for reductive groups *G* such that *G*(R) has discrete series to include characteristic *p* points at the boundary of weight space. In order to perform this construction, we define a notion of "locally analytic" functions and distributions on a locally Q_{p}-analytic manifold taking values in a complete Tate Z_{p}-algebra in which *p* is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on *p*-adic Lie groups given by Johansson and Newton.

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, 2018 Murray State University

#### Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

*Murray State Theses and Dissertations*

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ﬁlling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with ...

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

#### An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime 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.

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 ...

Parametric Polynomials For Small Galois Groups, 2018 Colby College

#### Parametric Polynomials For Small Galois Groups, Claire Huang

*Honors Theses*

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field ...

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.

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 $L_{d,[x_k]}(x)$.

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 ...

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.