Portland State University
University of New Mexico
California State University, San Bernardino
New Mexico State University
Montclair State University
Benedictine University
Eastern Illinois University
On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields,
2023
The Graduate Center, City University of New York
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.
Lagrange’S Study Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Study Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of Wilson’S Theorem—And More!,
2023
Ursinus College
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,
2023
Ursinus College
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,
2023
Ursinus College
Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem,
2023
Case Western Reserve University
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
Decomposition Of Beatty And Complementary Sequences,
2023
Smith College
Decomposition Of Beatty And Complementary Sequences, Geremías Polanco
Mathematics and Statistics: Faculty Publications
In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves,
2023
University of Tennessee, Knoxville
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein
Doctoral Dissertations
Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …
Coefficients Of Modular Forms And Applications To Partition Theory,
2023
University of Tennessee
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Doctoral Dissertations
We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane,
2023
Bellarmine University
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane, Kathleen Zopff
Undergraduate Theses
P-adic numbers are numbers valued by their divisibility by high powers of some prime, p. These numbers are an important concept in number theory that are used in major ideas such as the Reimann Hypothesis and Andrew Wiles’ proof of Fermat’s last theorem, and also have applications in cryptography. In this project, we will explore various visualizations of p-adic numbers. In particular, we will look at a mapping of p-adic numbers into the real plane which constructs a fractal similar to a Sierpinski p-gon. We discuss the properties of this map and give formulas for the sharp bounds of its …
Number Theoretic Arithmetic Functions And Dirichlet Series,
2023
CUNY New York City College of Technology
Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov
Publications and Research
In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.
Structure Of Extremal Unit Distance Graphs,
2023
University of South Carolina - Columbia
Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon
Senior Theses
This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.
Ramanujan–Sato Series For 1/Π,
2023
The University of Texas Rio Grande Valley
Ramanujan–Sato Series For 1/Π, Timothy Huber, Daniel Schultz, Dongxi Ye
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
We compute Ramanujan–Sato series systematically in terms of Thompson series and their modular equations. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof,
2023
Liberty University
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren
Senior Honors Theses
It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …
Euler Archive Spotlight,
2023
University of Washington, Tacoma
Euler Archive Spotlight, Erik R. Tou
Euleriana
A survey of two translations posted to the Euler Archive in 2022.
Using Bloom's Taxonomy For Math Outreach Within And Outside The Classroom,
2023
Benedictine University
Using Bloom's Taxonomy For Math Outreach Within And Outside The Classroom, Manmohan Kaur
Journal of Humanistic Mathematics
Not everyone is a great artist, but we don’t often hear, “I dislike art.” Most people are able to appreciate visual arts, music and sports, without necessarily excelling in it themselves. On the other hand, the phrase “I dislike math” is widely prevalent. This is especially ironic in our current society, where mathematics affects our day-to-day activities in essential ways such as e-commerce and e-mail. This paper describes the opportunity to popularize mathematics by focusing on its fun and creative aspects, and illustrates this opportunity through a brief discussion of interdisciplinary topics that expose the beauty, elegance and value of …
A Visual Tour Of Dynamical Systems On Color Space,
2023
Claremont Colleges
A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman
HMC Senior Theses
We can think of a pixel as a particle in three dimensional space, where its x, y and z coordinates correspond to its level of red, green, and blue, respectively. Just as a particle’s motion is guided by physical rules like gravity, we can construct rules to guide a pixel’s motion through color space. We can develop striking visuals by applying these rules, called dynamical systems, onto images using animation engines. This project explores a number of these systems while exposing the underlying algebraic structure of color space. We also build and demonstrate a Visual DJ circuit board for …
Generalized Far-Difference Representations,
2023
Claremont Colleges
Generalized Far-Difference Representations, Prakod Ngamlamai
HMC Senior Theses
Integers are often represented as a base-$b$ representation by the sum $\sum c_ib^i$. Lekkerkerker and Zeckendorf later provided the rules for representing integers as the sum of Fibonacci numbers. Hannah Alpert then introduced the far-difference representation by providing rules for writing an integer with both positive and negative multiples of Fibonacci numbers. Our work aims to generalize her work to a broader family of linear recurrences. To do so, we describe desired properties of the representations, such as lexicographic ordering, and provide a family of algorithms for each linear recurrence that generate unique representations for any integer. We then prove …
Elliptic Curves Over Finite Fields,
2023
Colby College
Elliptic Curves Over Finite Fields, Christopher S. Calger
Honors Theses
The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …
Elliptic Functions And Iterative Algorithms For Π,
2023
University of North Florida
Elliptic Functions And Iterative Algorithms For Π, Eduardo Jose Evans
UNF Graduate Theses and Dissertations
Preliminary identities in the theory of basic hypergeometric series, or `q-series', are proven. These include q-analogues of the exponential function, which lead to a fairly simple proof of Jacobi's celebrated triple product identity due to Andrews. The Dedekind eta function is introduced and a few identities of it derived. Euler's pentagonal number theorem is shown as a special case of Ramanujan's theta function and Watson's quintuple product identity is proved in a manner given by Carlitz and Subbarao. The Jacobian theta functions are introduced as special kinds of basic hypergeometric series and various relations between them derived using the triple …
