On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, 2022 University of Rochester

#### On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

*Rose-Hulman Undergraduate Mathematics Journal*

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of ...

An Overview Of Monstrous Moonshine, 2022 Cedarville University

#### An Overview Of Monstrous Moonshine, Catherine E. Riley

*Channels: Where Disciplines Meet*

The Conway-Norton monstrous moonshine conjecture set off a quest to discover the connection between the Monster and the *J*-function. The goal of this paper is to give an overview of the components of the conjecture, the conjecture itself, and some of the ideas that led to its solution. Special focus is given to Klein's *J*-function.

John Horton Conway: The Man And His Knot Theory, 2022 East Tennessee State University

#### John Horton Conway: The Man And His Knot Theory, Dillon Ketron

*Electronic Theses and Dissertations*

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

The Examination Of The Arithmetic Surface (3, 5) Over Q, 2022 California State University - San Bernardino

#### The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

*Electronic Theses, Projects, and Dissertations*

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i^{2} = 3, j^{2} = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL_{2}(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This list of ...

Symmetric Presentations Of Finite Groups And Related Topics, 2022 California State University - San Bernardino

#### Symmetric Presentations Of Finite Groups And Related Topics, Samar Mikhail Kasouha

*Electronic Theses, Projects, and Dissertations*

A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : (2 ...

A Study Of Semiprime Arithmetic Sequences, 2022 Belmont University

#### A Study Of Semiprime Arithmetic Sequences, Sam C. Uselton

*Honors Theses*

Semiprimes are positive integers of the form n = p*q where p and q are primes that are not necessarily distinct. Semiprimes can occur in arithmetic sequence, and the upper bound changes depending on the distance, d, between semiprimes in the sequence. The goal of this work is to present and prove theorems for different cases of semiprime arithmetic sequences, including d = 2k+1 where k is greater than or equal to 0 and d=2k where k is positive and not a multiple of 3, as well as a general case for finding upper bounds for varying distances. The ...

A Strange Attractor Of Primes, 2022 Ohio Northern University

#### A Strange Attractor Of Primes, Alexander Hare

*ONU Student Research Colloquium*

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding *k* terms (where *k* is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we ...

An Introduction To Number Theory, 2022 Portland State University

#### An Introduction To Number Theory, J. J. P. Veerman

*PDXOpen: Open Educational Resources*

These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed.

Chapters 1-14 represent almost 3 trimesters of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. The current content represents courses the author taught in the academic years 2020-2021 and 2021-2022.

It is a work in progress. If you have questions or comments, please contact Peter Veerman (veerman@pdx.edu).

Provably Weak Instances Of Plwe Revisited, Again, 2022 College of Saint Benedict/Saint John's University

#### Provably Weak Instances Of Plwe Revisited, Again, Katherine Mendel

*CSBSJU Distinguished Thesis*

Learning with Errors has emerged as a promising possibility for postquantum cryptography. Variants known as RLWE and PLWE have been shown to be more efficient, but the increased structure can leave them vulnerable to attacks for certain instantiations. This work aims to identify specific cases where proposed cryptographic schemes based on PLWE work particularly poorly under a specific attack.

Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, 2022 New Jersey City University

#### Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, Gunhan Caglayan

*Journal of Humanistic Mathematics*

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs.

Generating B-Nomial Numbers, 2022 Shippensburg University of Pennsylvania

#### Generating B-Nomial Numbers, Ji Young Choi

*Communications on Number Theory and Combinatorial Theory*

This paper presents three new ways to generate each type of b-nomial numbers: We develop ordinary generating functions, we find a whole new set of recurrence relations, and we identify each *b*-nomial number as a single binomial coefficient or as an alternating sum of products of two binomial coefficients.

A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, 2021 University of Maryland, College Park

#### A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn

*Rose-Hulman Undergraduate Mathematics Journal*

Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary ...

A Study In Applications Of Continued Fractions, 2021 California State University, San Bernardino

#### A Study In Applications Of Continued Fractions, Karen Lynn Parrish

*Electronic Theses, Projects, and Dissertations*

This is an expository study of continued fractions collecting ideas from several different sources including textbooks and journal articles. This study focuses on several applications of continued fractions from a variety of levels and fields of mathematics. Studies begin with looking at a number of properties that pertain to continued fractions and then move on to show how applications of continued fractions is relevant to high school level mathematics including approximating irrational numbers and developing new ideas for understanding and solving quadratics equations. Focus then continues to more advanced applications such as those used in the studies of number theory ...

Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, 2021 Louisiana State University and Agricultural and Mechanical College

#### Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, Shashika Petta Mestrige

*LSU Doctoral Dissertations*

Ramanujan in $1920$s discovered remarkable congruence properties of the partition function $p(n)$. Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and Hughes extended these works to $k$-colored partition functions. In $2010$, Folsom-Kent-Ono and Boylan-Webb proved the congruences of $p(n)$ by studying a $\ell$-adic module associated with a certain sequence of modular functions which are related to $p(n)$.

Primary goal of this thesis is to generalize the work of Atkin, Gordon, Hughes, Folsom-Kent-Ono, and Boylan-Webb about the partition function to a larger class of partition functions. For this ...

Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, 2021 The University of Western Ontairo

#### Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos

*Electronic Thesis and Dissertation Repository*

Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$-torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with non-trivial $p$-torsion by the $\F_p$-rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups ...

Introduction To Discrete Mathematics: An Oer For Ma-471, 2021 CUNY Queensborough Community College

#### Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas

*Open Educational Resources*

The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore **logics**, both propositional and first-order , and the construction of **proofs**, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through **set theory**. This theory is then put in practice in several applications. The particular (but quite widespread) case of **equivalence and order relations** is studied with detail. Then we introduces **sequences and proofs by induction**, followed by **number theory**. Finally, a small introduction to **combinatorics** is given.

Genus Bounds For Some Dynatomic Modular Curves, 2021 The University of Western Ontario

#### Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring

*Electronic Thesis and Dissertation Repository*

We prove that for every $n \ge 10$ there are at most finitely many values $c \in \mathbb{Q} $ such that the quadratic polynomial $x^2 + c$ has a point $\alpha \in \mathbb{Q} $ of period $n$. We achieve this by proving that for these values of $n$, every $n$-th dynatomic modular curve has genus at least two.

Contemporary Mathematical Approaches To Computability Theory, 2021 Western University

#### Contemporary Mathematical Approaches To Computability Theory, Luis Guilherme Mazzali De Almeida

*Undergraduate Student Research Internships Conference*

In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous.

Elliptic Curves And Their Practical Applications, 2021 Missouri State University

#### Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv

*MSU Graduate Theses*

Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.

Contributions To The Teaching And Learning Of Fluid Mechanics, 2021 Montclair State University

#### Contributions To The Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

*Department of Mathematics Facuty Scholarship and Creative Works*

This issue showcases a compilation of papers on fluid mechanics (FM) education, covering different sub topics of the subject. The success of the first volume [1] prompted us to consider another follow-up special issue on the topic, which has also been very successful in garnering an impressive variety of submissions. As a classical branch of science, the beauty and complexity of fluid dynamics cannot be overemphasized. This is an extremely well-studied subject which has now become a significant component of several major scientific disciplines ranging from aerospace engineering, astrophysics, atmospheric science (including climate modeling), biological and biomedical science and engineering ...