Number Theory Commons^™

Prime Factorization And Unit Calculations Of Quadratic Integer Rings, Gabriel F. Roca

Honors Undergraduate Theses

The failure of unique factorization in a ring leads to the investigation of the closest algebraic structure, which are prime ideals. Using generalizations that have helped solve questions such as Fermat's Last Theorem, there is interest to study the elements with a multiplicative inverse (units) via the geometry and arithmetic patterns that arise in quadratic integer rings, since they provide tools for other questions in mathematics, ranging from pure algebra to applications in cryptography, and more. Overall, the following thesis provides a small exposition on the theory of integral domains and some specific calculations.

Noncrystallographic Tail-Triangle C-Groups Of Rank 4 And Interlacing Number 2, Mark L. Loyola, Nonie Elvin S. Leyrita, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and k = 3,4,5, or 6. As moduli, we use the primes in the quadratic integer ring Z[τ], where τ = (1+√5)/2, the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable.

Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

Dissertations, Theses, and Capstone Projects

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

Diagrams In Involutive Residuated Lattices, Isis A. Gallardo

Electronic Theses and Dissertations

First, we show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thereby improving the existing Cayley/Holland style embedding theorem. Using this result, we demonstrate that the variety of all dis tributive lattice-ordered pregroups is generated by the functional algebra on the integers. Additionally, we prove that the equational theory of this variety is decidable.

Next, we establish that DLP is equal to the join of its subvarieties LP_n, where 𝑛 ∈ ℤ⁺, consisting of 𝑛-periodic ℓ-pregroups. We also prove that every algebra in LP_n can be embedded …

Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles

All Dissertations

This dissertation will explore factorization within orders in a number ring. By far the most well-understood of these orders are rings of algebraic integers. We will begin by examining how certain types of subrings may relate to the larger rings in which they are contained. We will then apply this knowledge, along with additional techniques, to determine how the elasticity in an order relates to the elasticity of the full ring of algebraic integers. Using many of the same strategies, we will develop a corresponding result in the rings of formal power series. Finally, we will explore a number of …

Some Experiments In Additive Number Theory, Yunan Wang

All Dissertations

This dissertation explores fundamental conjectures in number theory, focusing on the distribution patterns of representation functions in prime pairs. The work concentrates on twin primes, cousin primes, and primes separated by six units, offering a fresh heuristic interpretation of the Hardy-Littlewood correction factor. The analysis progresses to investigate the partition function for prime pairs in the form $(p, p+k)$, specifically for $k = 2, 4, 6$. The study culminates in the derivation of a general formula for prime pairs $(p, p+d)$, where $d$ is an even integer. Drawing on the insights gleaned from examining the correction factor, this dissertation proposes …

Bivariate Polynomials Of Low Degree And Small Mahler Measure, Souad El Otmani

BAU Journal - Science and Technology

In this work, we highlight that many of the known limit points of the Mahler measure of univariate polynomials can be obtained as the Mahler measure of low-degree bivariate polynomials. To this end, we provide for each relevant measure the corresponding original bivariate polynomial found in the literature, along with the corresponding low-degree polynomial with an analogous measure.

Combinatorial Problems On The Integers: Colorings, Games, And Permutations, Collier Gaiser

Electronic Theses and Dissertations

This dissertation consists of several combinatorial problems on the integers. These problems fit inside the areas of extremal combinatorics and enumerative combinatorics.

We first study monochromatic solutions to equations when integers are colored with finitely many colors in Chapter 2. By looking at subsets of {1, 2, . . . , n} whose least common multiple is small, we improved a result of Brown and Rödl on the smallest integer n such that every 2-coloring of {1, 2, . . . , n} has a monochromatic solution to equations with unit fractions. Using a recent result of Boza, …

Bifurcations And Resultants For Rational Maps And Dynatomic Modular Curves In Positive Characteristic, Colette Lapointe

Dissertations, Theses, and Capstone Projects

No abstract provided.

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

Dissertations, Theses, and Capstone Projects

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, Brad Isaacson

Publications and Research

In a previous paper, we expressed three families of character sums by certain generalized Bernoulli functions which in turn were expressed by generalized Bernoulli numbers via a complicated and indirect process. In this paper, we generalize a theorem of Ibukiyama to directly express these generalized Bernoulli functions by generalized Bernoulli numbers. As a result, we can express the three families of character sums by generalized Bernoulli numbers in a more elegant fashion than was done before.

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

Departmental Honors & Graduate Capstone Projects

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

Murmurations And Root Numbers, Alexey Pozdnyakov

University Scholar Projects

We report on a machine learning investigation of large datasets of elliptic curves and L-functions. This leads to the discovery of murmurations, an unexpected correlation between the root numbers and Dirichlet coefficients of L-functions. We provide a formal definition of murmurations, describe the connection with 1-level density, and provide three examples for which the murmuration phenomenon has been rigorously proven. Using our understanding of murmurations, we then build new machine learning models in search of a polynomial time algorithm for predicting root numbers. Based on our models and several heuristic arguments, we conclude that it is unlikely for …

Hilbert Reciprocity Over Number Fields, Dillon Snyder

Honors Scholar Theses

A Hilbert symbol has the value 1 or −1 depending on the existence of solutions to a certain quadratic equation in a local field, R, or C. Hilbert reciprocity states that for a number field F and two nonzero a and b in F, the product of Hilbert symbols associated to a and b at all the places of F is 1. That is, these Hilbert symbols are −1 for a finite, even number of places of F . Hilbert reciprocity when F = Q is equivalent to the classical quadratic reciprocity law, so Hilbert reciprocity in number fields can …

Rsa Algorithm, Evalisbeth Garcia Diazbarriga

ATU Scholars Symposium

I will be presenting about the RSA method in cryptology which is the coding and decoding of messages. My research will focus on proving that the method works and how it is used to communicate secretly.

Finite Monodromy And Artin Representations, Emma Lien

LSU Doctoral Dissertations

Artin representations, which are complex representations of finite Galois groups, appear in many contexts in number theory. The Langlands program predicts that Galois representations like these should arise from automorphic representations and many examples of this correspondence have been found such as in the proof of Fermat's Last Theorem. This dissertation aims to make an analysis of explicitly computable examples of Artin representations from both sides of this correspondence. On the automorphic side, certain weight 1 modular forms have been shown to be related to Artin representations and an explicit analysis of their Fourier coefficients allows us to identify the …

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

Rose-Hulman Undergraduate Mathematics Journal

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, Krishna Rajesh

HMC Senior Theses

Ron Graham's sequence is a surprising bijection from the natural numbers to the non-prime integers, which is constructed by looking at sequences whose product is square. In this thesis we will resolve a 22-year-old conjecture about this bijection, by construction of explicit sequences in a modified number theoretic context. Additionally, we will discuss the history of this problem, and give computational techniques for computing this bijection, levering ideas from linear algebra over the finite field of two elements.

Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, Danzhe Chen

CMC Senior Theses

This thesis provides a comprehensive exploration of lattice theory, emphasizing its dual significance in both theoretical mathematics and practical applications, particularly within computational complexity and cryptography. The study begins with an in-depth examination of the fundamental properties of lattices and progresses to intricate lattice-based problems such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems are analyzed for their computational depth and linked to the Subset Sum Problem (SSP) to highlight their critical roles in understanding computational hardness. The narrative then transitions to the practical applications of these theories in cryptography, evaluating the shift from …