Number Theory Commons^™

On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty

Mathematics Faculty Publications

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.

Drawing Numbers And Listening To Patterns, Loren Zo Haynes

Honors College Theses

The triangular numbers is a series of number that add the natural numbers. Parabolic shapes emerge when this series is placed on a lattice, or imposed with a limited number of columns that causes the sequence to continue on the next row when it has reached the kth column. We examine these patterns and construct proofs that explain their behavior. We build off of this to see what happens to the patterns when there is not a limited number of columns, and we formulate the graphs as musical patterns on a staff, using each column as a line or space …

General Multi-Sum Transformations And Some Implications, James Mclaughlin

Mathematics Faculty Publications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence {g(k)}), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials, and various multisums that are expressible as infinite products.

The History And Applications Of Fibonacci Numbers, Cashous W. Bortner, Allan C. Peterson

UCARE: Research Products

The Fibonacci sequence is arguably the most observed sequence not only in mathematics, but also in nature. As we begin to learn more and more about the Fibonacci sequence and the numbers that make the sequence, many new and interesting applications of the have risen from different areas of algebra to market trading strategies. This poster analyzes not only the history of Leonardo Bonacci, but also the elegant sequence that is now his namesake and its appearance in nature as well as some of its current mathematical and non-mathematical applications.

Cyclic Critical Groups Of Graphs, Ryan P. Becker, Darren B. Glass

Math Faculty Publications

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly.

Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty

Mathematics Faculty Publications

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than ℤ. We then study the growth of the p^∞- Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers k ⊂ K ⊂ F in which [F : K] is not a p-power extension.

Visual Properties Of Generalized Kloosterman Sums, Paula Burkhardt '16, Alice Zhuo-Yu Chan '14, Gabriel Currier '16, Stephan Ramon Garcia, Florian Luca, Hong Suh '16

Pomona Faculty Publications and Research

For a positive integer m and a subgroup A of the unit group (Z/mZ)^x, the corresponding generalized Kloosterman sum is the function K(a, b, m, A) = Σ_uEA e(au+bu^-1/m). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

Adinkras And Arithmetical Graphs, Madeleine Weinstein

HMC Senior Theses

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding …

A Computational And Theoretical Exploration Of The St. Petersburg Paradox, Alexander Olivero

Undergraduate Honors Thesis Collection

This thesis displays a sample distribution, generated from both a simulation (for large n) by computer program and explicitly calculated (for smaller n), that is not governed by the Central Limit Theorem and, in fact seems to display chaotic behavior. To our knowledge, the explicit calculation of the sample distribution function is new. This project outlines the results that have found a relation to number theory in a probabilistic game that has perplexed mathematicians for hundreds of years.

A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, Russell Lee Jahn

All Graduate Theses, Dissertations, and Other Capstone Projects

In this paper we demonstrate how the principles of measure theory can be applied effectively to problems of number theory. Initially, necessary concepts from number theory will be presented. Next, we state standard concepts and results from measure theory to which we will need to refer. We then develop our repertoire of measure theoretic machinery by constructing the needed measures and defining a generalized version of the multiplicative convolution of measures. A suitable integration by parts formula, one that is general enough to handle various combinations of measures, will then be derived. At this juncture we will be ready to …

Refinements Of Some Partition Inequalities, James Mclaughlin

Mathematics Faculty Publications

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if M ≥ 5 is an integer and the integers a and b are relatively prime to M and satisfy 1 ≤ a < b < M/2, and the c(m, n) are defined by 1 (sqa, sqM−a; qM)∞ − 1 (sqb , sqM−b ; qM)∞ := X m,n≥0 c(m, n)s mq n , then c(m, Mn) ≥ 0 for all integers m ≥ 0, n ≥ 0. A similar result is proved for the integers d(m, n) defined by (−sqa , −sqM−a ; q M)∞ − (−sqb , −sqM−b ; q M)∞ := X m,n≥0 d(m, n)s mq n . In each case there are obvious interpretations in terms of integer partitions. For example, if p1,5(m, n) (respectively p2,5(m, n)) denotes the number of partitions of n into exactly m parts ≡ ±1( mod 5) (respectively ≡ ±2( mod 5)), then for each integer n ≥ 1, p1,5(m, 5n) ≥ p2,5(m, 5n), 1 ≤ m ≤ 5n.

Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, Jonathan A. Constable

Theses and Dissertations--Mathematics

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.

In the second chapter we introduce the class number, proper class number and complete class number as …

The Monochromatic Column Problem: The Prime Case, Loran Elizabeth Crowell

Online Theses and Dissertations

Let p1, p2, . . . , pn be pairwise coprime positive integers and let P = p1p2 · · · pn. Let 0,1,...,m−1 be a sequence of m different colors. Let A be an n×mP matrix of colors in which row i consists of blocks of pi consecutive entries of the same color, with colors 0 through m − 1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A in which every entry is the same color. A partial solution for the case when m is prime is given.

Automated Conjecturing Approach To The Discrete Riemann Hypothesis, Alexander Bradford

Theses and Dissertations

This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious" function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz's Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded.

A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot

College of Graduate Studies: Theses & Dissertations

Nearly a century ago, the mathematicians Hardy and Ramanujan established their celebrated circle method to give a remarkable asymptotic expression for the unrestricted partition function. Following later improvements by Rademacher, the method was utilized by Niven, Lehner, Iseki, and others to develop rapidly convergent series representations of various restricted partition functions. Following in this tradition, we use the circle method to develop formulas for counting the restricted classes of partitions that arise in the Gollnitz-Gordon identities. We then show that our results are strongly supported by numerical tests. As a side note, we also derive and compare the asymptotic behavior …

An Exposition Of The Eisenstein Integers, Sarada Bandara

Masters Theses

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[ω] = {a+bω ∣ a, b ∈ ℤ} where ω² + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bi ∣ a, b ∈ …

Tabulating Pseudoprimes And Tabulating Liars, Andrew Shallue

Scholarship

This paper explores the asymptotic complexity of two problems related to the Miller-Rabin-Selfridge primality test. The first problem is to tabulate strong pseudoprimes to a single fixed base $a$. It is now proven that tabulating up to $x$ requires $O(x)$ arithmetic operations and $O(x\log{x})$ bits of space. The second problem is to find all strong liars and witnesses, given a fixed odd composite $n$. This appears to be unstudied, and a randomized algorithm is presented that requires an expected $O((\log{n})^2 + |S(n)|)$ operations (here $S(n)$ is the set of strong liars). Although interesting in their own right, a notable application …

Applying Dijkstra Algorithm For Solving Neutrosophic Shortest Path Problem, Florentin Smarandache, Luige Vladareanu, Said Broumi, Assia Bakali, Muhammad Akram

Branch Mathematics and Statistics Faculty and Staff Publications

The selection of shortest path problem is one the classic problems in graph theory. In literature, many algorithms have been developed to provide a solution for shortest path problem in a network. One of common algorithms in solving shortest path problem is Dijkstra’s algorithm. In this paper, Dijkstra’s algorithm has been redesigned to handle the case in which most of parameters of a network are uncertain and given in terms of neutrosophic numbers. Finally, a numerical example is given to explain the proposed algorithm.

Luhn Primes Of Order Ω, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

A prime p, that added to its reverse results in a new prime is called a Luhn prime . The number 229 is a Luhn prime because 229 + 922 = 1151 and 1151 is also a prime . We extend the definition of Luhn primes, present some of the of their properties and ways to determine them .

Single Valued Neutrosophic Graphs: Degree, Order And Size, Florentin Smarandache, Said Broumi, Mohamed Talea, Assia Bakali

Branch Mathematics and Statistics Faculty and Staff Publications

The single valued neutrosophic graph is a new version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.