Number Theory Commons^™

From Counting To Advanced Math: How Five Simple Axioms Shape The Mathematical Landscape, Joy Skaggs

ATU Scholars Symposium

In the late nineteenth and early twentieth centuries, mathematics faced a foundational crisis: Greog Cantor’s set theory led to an interesting self-referencing paradox in math and questions about the logical consistency of mathematics. From this crisis, two opposing viewpoints emerged: the Formalists and the Intuitionists. The Formalists praised Cantor’s work as a way to place math on a secure logical foundation, ensuring the discipline’s purity, however the Intuitionists despised Cantor’s work and heralded Cantor as a charlatan and corrupter of the youth. The leader of the Formalists, David Hilbert, proposed a formal system of rigorous proofs to build a complete …

The Modern History Of The Basel Problem, John Campbell, Paul Levrie

Euleriana

The \emph{Basel problem} refers to the problem of determining a closed form for the infinite series $\frac{1}{1^2} + \frac{1}{2^2} + \cdots$. If we consider what mathematical results have the most peer-reviewed papers devoted to new ways of proving such results, Euler's formula $\frac{1}{1^2} + \frac{1}{2^2} + \cdots = \frac{\pi^2}{6}$ is certainly among the top of such results. This motivates our historical study of peer-reviewed papers based on proofs of Euler's formula, and we introduce what appears to be the most comprehensive and up-to-date and exhaustive catalogue of peer-reviewed, published papers in the 20th and 21st centuries devoted to or mainly …

A Virtual Community Math Circle, Skona Brittain, Sayonita Ghosh Hajra, Steve Heller, Daniel Hodgins, Peter Petto, Gabriella Pinter, Lauren Rose, A. Gwinn Royal, Asmita Sodhi

Journal of Math Circles

The Julia Robinson Mathematics Festival (JRMF) Community Math Circle is free, volunteer-run, and online. We collaborate with JRMF and use their activities in our events. Started during the pandemic, the Community Math Circle continues to thrive. We have about 40 participants attending every month, typically kids aged 6 to 13, teachers, facilitators, and other adults. This article describes how our event is organized, planned, and executed, and how we train facilitators. We will also offer reflections on our successes and challenges.

Polygonal Number Similarity, Gunhan Caglayan

Journal of Humanistic Mathematics

This note takes an exploratory approach to define and then visualize the notion of polygonal number similarity between pairs of k-gonal numbers P_k(αn) and P_k(n) , where α is an integer scale factor of 2 or greater.

The Interaction Between Additive And Multiplicative Structures In Arithmetic Settings, Ali Alsetri

Theses and Dissertations--Mathematics

The first part of this thesis is concerned with Goldbach-type problems. In recent years, there has been an interest in developing density versions of Goldbach-type results. Namely, given a relatively dense subset A of the primes, one may study representations of integers as sums of primes belonging to the subset A. These density Goldbach-type results have been facilitated by the development of new tools from additive combinatorics, in particular the Fourier-analytic transference principle due to Green. We apply the transference principle to obtain a variant of Vinogradov’s theorem involving subsets of primes confined to the residue class 1 (mod 3). …

A New Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

Discussion Of The Collatz Conjecture, Morgan Hayes

Student Scholar Symposium

The Collatz conjecture was introduced in 1937, but, despite the efforts of many mathematicians, it has yet to be proven. It appears to have very little connection to other areas of mathematics, although some aspects of it allude to questions similar to those asked in studies of prime numbers and the Riemann Hypothesis. The nature of the Collatz sequence allows it to be useful in cryptography and semi-random number generation. Some, including Craig Alan Feinstein, believe a formal proof of the conjecture is not possible. Others believe mathematics has not been developed enough to resolve the issue. To enter into …

(R2094) On Mds Bisymmetric Rhotrices Using Self-Dual Bases And Conjugate Elements Of Finite Fields, Shalini Gupta, Ruchi Narang, Manpreet Singh

Applications and Applied Mathematics: An International Journal (AAM)

Bisymmetric matrices have wide range of applications in statistics, engineering problems, information theory and computer science including coding theory and cryptography. In cryptography, a rhotrix being a couple matrix doubles the security of the cryptosystem. Here, we construct maximum distance separable (MDS) bisymmetric rhotrices using self-dual bases and conjugate elements of finite fields. MDS rhotrices are very crucial for the designing of block ciphers and hash functions in cryptography.

(Si15-022) Five Efficient Cryptography Authentication Schemes With Functional Relation On General Spaces, L. Sreenivasulu Reddy, S. Lakshmisri

Applications and Applied Mathematics: An International Journal (AAM)

This work addresses the growing demand for diversification in cryptographic schemes to secure communication. This work proposes a novel suite of algorithms, including two block ciphers (TPBlock and TAP-Block), two stream ciphers (TP-Stream and TAP-Stream), and a zero-knowledge proof scheme (F-zero knowledge proof). All schemes leverage functional relations defined over the real number space with a dimension greater than one for encryption, decryption, and key generation, offering an alternative to the number-theoretical aspects and algebraic structures commonly used in existing schemes. The main goal of this work is to introduce and propose these five novel cryptographic schemes to provide authentication …

(Si15-064) Permutation Pentanomials Over Finite Fields With Even Characteristic, Shalini Gupta, Sushil Kumar, Ashima .

Applications and Applied Mathematics: An International Journal (AAM)

Permutation polynomials over finite fields constitute an active area of research and play an important role in diverse domains, including finite geometry, combinatorial design, coding theory, and cryptography. The study of these polynomials has a long history, and many results have been obtained in recent years. This paper presents new classes of permutation pentanomials based on permutation over the unit circle of finite fields with even characteristic that contribute to the theoretical development of permutation polynomials.

Incremental Increases Between Successive Integers When Raised To The Nth Power, Sutton J. Olesen

Rose-Hulman Undergraduate Mathematics Journal

For thousands of years, the beautiful field of number theory has captivated mathematicians with its elegant simplicity. Positive integers continue to reveal properties and relationships that are a joy to uncover, and in this paper, we investigate a pattern involving exponents and factorials while exploring some common notations in the field of number theory. Combinatorics, the field dealing with the mathematics of counting and arranging, also holds a presence in this paper. Pascal’s Triangle–the foundation of binomial expressions, also comes into play due to its tight relationship with combinatorics. Pascal’s Identity, the property that builds the triangle, becomes very useful …

An Interpolating Regime For The Extreme Values Of A Random Zeta Model, Christine K. Chang

Dissertations, Theses, and Capstone Projects

We study the large values of a random model of the Riemann zeta function over short intervals. The extreme value statistics depend on the interval size: the log-correlated regime governs intervals of order one while the i.i.d. regime emerges over longer intervals. The main focus is to describe the transition between these two well-understood regimes as the interval varies in length. This thesis shows that there is an intermediate regime where the behavior of the zeta model’s maxima cannot be entirely captured by either extreme— i.i.d. or fully log-correlated. This suggests that the Riemann zeta function exhibits correlations around its …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses

Master's Theses

This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …

Parametrization Of Formal Norm Compatible Sequences, Joseph Dicapua

Dissertations, Theses, and Capstone Projects

We give a classification of power series parametrizing Lubin-Tate trace compatible sequences. This proof answers a question posed in the literature by Berger and Fourquaux. Lubin-Tate trace compatible sequences are a generalization of norm compatible sequences, which arise in Iwasawa theory and local class field theory. The result we prove generalizes the interpolation theorem proved by Coleman in the classical norm compatible sequence case. We also, jointly with Victor Kolyvagin, give a method for finding such series explicitly in certain special cases.

Existence And Nonexistence Of Positive Solutions For Fractional Boundary Value Problems With Lidstone-Inspired Fractional Conditions, Jeffrey Lyons, Jeffrey T. Neugebauer, Aaron G. Wingo

EKU Faculty and Staff Scholarship

This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann–Liouville fractional boundary value problems of order 𝛼 +2⁢𝑛, where 𝛼 ∈(𝑚 −1,𝑚] with 𝑚 ≥3 and 𝑚,𝑛 ∈ℕ. The conjugate fractional boundary conditions are inspired by Lidstone conditions. The nonlinearity depends on a positive parameter on which we identify constraints that determine the existence or nonexistence of positive solutions. Our method involves constructing Green’s function by convolving the Green functions of a lower-order fractional boundary value problem and a conjugate boundary value problem and using properties of this Green function to apply the Guo–Krasnosel’skii …

A Study Of The Sum Of Divisors, Henry M. Willie

Miners Solving for Tomorrow Research Conference

No abstract provided.

Patterns Within The Collatz Conjecture, Kiel Harrison

SACAD: Scholarly Activities

The Collatz Conjecture, also known as 3n+1, one of the most famous unsolved problems in mathematics, has been forever out of reach of being truly solved. However, through the application of traces, there is now a new pathway forward to working out a potential solution. This study shows how this pathway was found, and what steps need to be taken to follow it.

Algorithms For Order Statistics In Farey Sequences: A Computational Study, Connor Weyers

School of Computing: Dissertations, Theses, and Student Research

Farey sequences are the sets of irreducible fractions in increasing order with denominator less or equal to some integer n. They are a well-known concept in number theory problems and are related to many other concepts in number theory including integer factoring, Fibonacci sequences, and Riemann’s Zeta function. In this paper, we investigate some known algorithms to solve certain problems in Farey sequences from a computational perspective. In particular, we implement established algorithms that have not been previously implemented with the goal of creating a package that can be used more broadly. We also develop a new algorithm for rational …

Explicit Modularity And Moments For Certain Hypergeometric Character Sums, Brian Grove

LSU Doctoral Dissertations

A great deal of progress in number theory throughout history has been motivated by trying to solve equations. One of the most famous challenges is to show there are no positive integer solutions to $x^{n}+y^{n} = z^{n}$ for $n > 2$, posed by Fermat around 1637. Special cases, such as the $n = 3$ and $n = 4$ cases, can be established using various algebraic manipulations. However, a general solution was elusive until the late 1990s when the combined work of Wiles \cite{Wiles} and Taylor--Wiles \cite{TaylorWiles} give a full proof.

One of the key insights used in proving Fermat's conjecture involves …