Number Theory Commons^™

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.

Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan

Mathematics & Statistics Faculty Publications

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …

Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World, Dylan Phares

CMC Senior Theses

Shor's Algorithm is an extremely powerful tool, in utilizing this tool it is important to understand how it works and why it works. As well as the vast implications it could have for cryptography

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 …

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

Paley Graphs, Prime Graphs, And Crossword Puzzles, Robert D. Jacobs Jr.

Theses and Dissertations

In this paper, we will talk about many different mathematical concepts. We will prove theorems about Paley graphs, prime graphs, and crossword puzzles. It will be very fun.

The results in the section about Paley graphs include structure theorems about the subgraph induced by the quadratic residues, the subgraph induced by the non-residues and a few related subgraphs. The main is to better understand the “independence structure” of the Paley graph itself. No good upper bound on the independence number of Paley graphs is known. Theorems about these subgraphs, and various counts aim at future improvement of upper bounds for …

Zeckendorf Representation Analysis On Third Order Fibonacci Sequences That Do Not Satisfy The Uniqueness Property, Samuel A. Aguilar

Honors College Theses

Zeckendorf's Theorem states that every natural number can be expressed uniquely as the sum of distinct non-consecutive terms of the shifted Fibonacci sequence (i.e. 1, 2, 3, 5, ...). This theorem has motivated the study of representation of integers by the sum of non-adjacent terms of Nth order Fibonacci sequences, including the characterization of the uniqueness of Zeckendorf representation based on the initial terms of the sequence. Moreover, when this uniqueness property is satisfied for third order Fibonacci sequences, the ratio of integers less than a given number X that have a Zeckendorf representation has been estimated by Dr. Sungkon …

The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson

Cybersecurity Undergraduate Research Showcase

The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …

Further Generalizations Of Happy Numbers, E. Simonton Williams

Rose-Hulman Undergraduate Mathematics Journal

A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …

Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine

Rose-Hulman Undergraduate Mathematics Journal

We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.

Rough Numbers And Variations On The Erdős--Kac Theorem, Kai Fan

Dartmouth College Ph.D Dissertations

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory.

In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y>=1. Denoting by Phi(x,y) the count of these numbers up to any given x>=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)<0.6x/\log y holds uniformly for all 3<=y<=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges.

In the second part, we turn …

On The Order-Type Complexity Of Words, And Greedy Sidon Sets For Linear Forms, Yin Choi Cheng

Dissertations, Theses, and Capstone Projects

This work consists of two parts. In the first part, we study the order-type complexity of right-infinite words over a finite alphabet, which is defined to be the order types of the set of shifts of said words in lexicographical order. The set of shifts of any aperiodic morphic words whose first letter in the purely-morphic pre-image occurs at least twice in the pre-image has the same order type as Q ∩ (0, 1), Q ∩ (0, 1], or Q ∩ [0, 1). This includes all aperiodic purely-morphic binary words. The order types of uniform-morphic ternary words were also studied, …

On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields, Owen Sweeney

Dissertations, Theses, and Capstone Projects

We obtain a new simpler sufficient condition for Kolyvagin's criteria, regarding the second case of Fermat's last theorem with prime exponent p over the p-th cyclotomic field, to hold. It covers cases when the existing simpler sufficient conditions do not hold and is important for the theoretical study of the criteria.

On The Spectrum Of Quaquaversal Operators, Josiah Sugarman

Dissertations, Theses, and Capstone Projects

In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …

Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov

Publications and Research

The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.

Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson

All Dissertations

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …

Zeros Of Modular Forms, Daozhou Zhu

All Dissertations

Let $E_k(z)$ be the normalized Eisenstein series of weight $k$ for the full modular group $\text{SL}(2, \mathbb{Z})$. It is conjectured that all the zeros of the weight $k+\ell$ cusp form $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. Reitzes, Vulakh and Young \cite{Reitzes17} proved that this statement is true for sufficiently large $k$ and $\ell$. Xue and Zhu \cite{Xue} proved the cases when $\ell=4,6,8$ with $k\geq\ell$, all the zeros of $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ lie on the arc $|z|=1$. For all $k\geq\ell\geq10$, we will use the same method as \cite{Reitzes17} to locate these zeros on the arc $|z|=1$, and improve the …

The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.

One Formula For Non-Prime Numbers: Motivations And Characteristics, Mahmoud Mansour, Kamal Hassan Prof.

Basic Science Engineering

Primes are essential for computer encryption and cryptography, as they are fundamental units of whole numbers and are of the highest importance due to their mathematical qualities. However, identifying a pattern of primes is not easy. Thinking in a different way may get benefits, by considering the opposite side of the problem which means focusing on non-prime numbers. Recently, researchers introduced, the pattern of non-primes in two maximal sets while in this paper, non-primes are presented in one formula. Getting one-way formula for non-primes may pave the way for further applications based on the idea of primes.

Lagrange’S Study Of Wilson’S Theorem, Carl Lienert

Number Theory

No abstract provided.

Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert

Number Theory

No abstract provided.

Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert

Number Theory

No abstract provided.

Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert

Number Theory

No abstract provided.