Number Theory Commons^™

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.

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, 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 …

Meertens Number And Its Variations, Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Gödel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations of Meertens numbers where there is a finite time algorithm to decide whether such numbers exist, exhibit infinite families of these variations and provide bounds on parameters needed for their existence.

(R1979) Permanent Of Toeplitz-Hessenberg Matrices With Generalized Fibonacci And Lucas Entries, Hacène Belbachir, Amine Belkhir, Ihab-Eddine Djellas

Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, we evaluate the permanent and determinant of some Toeplitz-Hessenberg matrices with generalized Fibonacci and generalized Lucas numbers as entries.We develop identities involving sums of products of generalized Fibonacci numbers and generalized Lucas numbers with multinomial coefficients using the matrix structure, and then we present an application of the determinant of such matrices.

A Comparison Of Cryptographic Methods, Christopher Gilmore

Senior Honors Theses

While elliptic curve cryptography and quantum cryptography are significantly different branches of cryptography, they provide a suitable reference point for comparison of the value of developing methods used in the present and investing in methods to be used in the future. Elliptic curve cryptography is quite common today, as it is generally secure and efficient. However, as the field of cryptography advances, the value of quantum cryptography’s inherent security from its basic properties should be considered, as a fully realized quantum cryptosystem has the potential to be quite powerful. Ultimately, it is of critical importance to determine the value of …

(Si10-062) Comprehensive Study On Methodology Of Orthogonal Interleavers, Priyanka Agarwal, Shivani Dixit, M. Shukla, Gaurish Joshi

Applications and Applied Mathematics: An International Journal (AAM)

Interleaving permutes the data bits by employing a user defined sequence to reduce burst error which at times exceeds the minimum hamming distance. It serves as the sole medium to distinguish user data in the overlapping channel and is the heart of Interleave Division Multiple Access (IDMA) scheme. Versatility of interleavers relies on various design parameters such as orthogonality, correlation, latency and performance parameters like bit error rate (BER), memory occupancy and computation complexity. In this paper, a comprehensive study of interleaving phenomenon and discussion on numerous interleavers is presented. Also, the BER performance of interleavers using IDMA scheme is …

The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

Dissertations, Theses, and Capstone Projects

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …

Proving Dirichlet's Theorem On Arithmetic Progressions, Owen T. Abma

Undergraduate Student Research Internships Conference

First proved by German mathematician Dirichlet in 1837, this important theorem states that for coprime integers a, m, there are an infinite number of primes p such that p = a (mod m). This is one of many extensions of Euclid’s theorem that there are infinitely many prime numbers. In this paper, we will formulate a rather elegant proof of Dirichlet’s theorem using ideas from complex analysis and group theory.

Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau

Electronic Thesis and Dissertation Repository

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.

In 1985, Schoof devised an algorithm to compute the zeta function of an …

The Hfd Property In Orders Of A Number Field, Grant Moles

All Theses

We will examine orders R in a number field K. In particular, we will look at how the generalized class number of R relates to the class number of its integral closure R. We will then apply this to the case when K is a quadratic field to produce a more specific relation. After this, we will focus on orders R which are half-factorial domains (HFDs), in which the irreducible factorization of any element α∈R has fixed length. We will determine two cases in which R is an HFD if and only if its ring of …

Efficiency Of Homomorphic Encryption Schemes, Kyle Yates

All Theses

In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …

On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard

Doctoral Dissertations

We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding to …

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 either 0 …

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.

Structure Of Number Theoretic Graphs, Lee Trent

Mathematical Sciences Technical Reports (MSTR)

The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …

Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis

Student Research

While doing research looking for ways to categorize causal networks generated by Sequential Substitution Systems, I created a new notation to compactly summarize concatenations of integers or strings of integers, including infinite sequences of these, in the same way that sums, products, and unions of sets can be summarized. Using my method, any sequence of integers or strings of integers with a closed-form iterative pattern can be compactly summarized in just one line of mathematical notation, including graphs generated by Sequential Substitution Systems, many Primitive Pythagorean Triplets, and various Lucas sequences including the Fibonacci sequence and the sequence of square …

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.

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 : …

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 i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(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 …

A Study Of Semiprime Arithmetic Sequences, Sam C. Uselton

Honors Scholars Collaborative Projects

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 building …

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 introduce the functions “depth” – where depth(n) …

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, 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, 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, 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.

Hypergeometric Motives, David P. Roberts, Fernando Rodriguez Villegas

Mathematics Publications

No abstract provided.

Interpolating The Riemann Zeta Function In The P-Adics, Rebecca Mamlet

Scripps Senior Theses

In this thesis, we develop the Kubota-Leopoldt Riemann zeta function in the p-adic integers. We follow Neil Koblitz's interpolation of Riemann zeta, using Bernoulli measures and p-adic integrals. The underlying goal is to better understand p-adic expansions and computations. We finish by connecting the Riemann zeta function to L-functions and their p-adic interpolations.

An Integration Of Art And Mathematics, Henry Jaakola

Undergraduate Honors Theses

Mathematics and art are seemingly unrelated fields, requiring different skills and mindsets. Indeed, these disciplines may be difficult to understand for those not immersed in the field. Through art, math can be more relatable and understandable, and with math, art can be imbued with a different kind of order and structure. This project explores the intersection and integration of math and art, and culminates in a physical interdisciplinary product. Using the Padovan Sequence of numbers as a theoretical basis, two artworks are created with different media and designs, yielding unique results. Through these pieces, the order and beauty of number …