Number Theory Commons^™

(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.

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.

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. …

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

Miners Solving for Tomorrow Research Conference

No abstract provided.

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 …

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.

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 …

The Intricacies Of Pairwise Modular Multiplicative Inverse In Lucas Numbers, Charles Liu

Rose-Hulman Undergraduate Mathematics Journal

Let (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ < q, qq′ ≡ 1 (mod p), q′ < p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers.

Proportion Of P-Adic Polynomials Which Are Irreducible, Isaac Rajagopal

Rose-Hulman Undergraduate Mathematics Journal

We attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots.

Welcome 2025—The Year Of A Magical Number, Md Sadikur Rahman

Journal of Humanistic Mathematics

This is a poem to welcome the new year 2025 and note its relevance to mathematics.

The Frequency Of Elliptic Curves Over $\Mathbb{Q}[I]$ With Fixed Torsion, Alan S. Zhao

Rose-Hulman Undergraduate Mathematics Journal

Mazur\textsc{\char13}s Theorem states that there are precisely 15 possibilities for the torsion subgroup of an elliptic curve defined over the rational numbers. It was previously shown by Harron and Snowden that the number of isomorphism classes of elliptic curves of height up to $X$ that have a specific torsion subgroup $G$ is on the order of $X^{1/{d(G)}}$, for some positive $d(G)$ depending on $G$. We compute $d(G)$ for these groups over $\Qi$. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field $\Qi$. We compute the …

An Algorithm And Computation To Verify Legendre's Conjecture Up 7 · 1013, Jonathan Sorenson, Jonathan Webster

Computer Science and Software Engineering

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integer n, there exists a prime between n² and (n + 1)². Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between n2 and n(n + 1) and also between n(n + 1) and (n + 1)². Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all n ≤ N in time O(N log N log …

Galoistheory And The Arithmetic-Geometric Series, Daniel Vargas

HMC Senior Theses

Motivated by classical works of Gauss and Euler on the AGM, Ono and his

collaborators Griffin et al. (2023); McSpirit and Ono (2023) have investigated

the union of AGM sequences over finite fields 𝔽𝑞, where 𝑞 ≡3 mod 4. A

recent preprint Kayath et al. (2024) extends some of their results to all finite

fields with odd characteristic. We refine these works when 𝑞≡5 mod 8. In

particular, we explicitly determine the components of these graphs and their

total population. We also use Galois-theoretic results to make progress in

the search for cycles over finite fields with odd characteristic.

Strong Neighborhood-Prime Labelings Under Ordinary & Gaussian Integers, Micheal Arnal-Brown

Murray State Theses and Dissertations

This thesis introduces and studies the notion of a strong neighborhood-prime labeling, a strengthening of the neighborhood-prime labeling where the label 1 can be assigned to any vertex in a graph. We prove that several graph families—including paths, cycles (excluding those congruent to 2 modulo 4), caterpillars, helm graphs, closed helm graphs, gear graphs, and graphs with universal vertices—admit such labelings, and also provide results to more general classes of graphs. We extend this new labeling concept to the Gaussian integers using a spiral order- ing on Z[i] and define a Gaussian analogue of strongly neighborhood-primeness. To support this extension, …

The Combinatorics Of Integer Partitions Enumerated By Some Exotic Weights, Hunter Waldron

Dissertations, Master's Theses and Master's Reports

Identities of integer partitions generally state that two dissimilar appearing families of partitions are in fact equinumerous when both are restricted to any fixed size. Euler's theorem is a classic example of such an identity, which equates the number of partitions with odd parts to the number of partitions with distinct parts. Lately, analogs of known partition identities involving weights other than size have begun to attract research interest. This dissertation is an investigation of two such weights. In Chapter 2, we study Schmidt weights, which count only parts with indices belonging to some given subset of the positive integers. …