Polynomial Factoring Algorithms And Their Computational Complexity, 2014 University of Connecticut - Storrs

#### Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna

*Honors Scholar Theses*

Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to ...

Fermat's Last Theorem, 2014 Lake Forest College

#### Fermat's Last Theorem, William Forcier

*Senior Theses*

First conjectured by Fermat in the 1630s, Fermat's Last Theorem has cause a great deal of advancement in the field of number theory. It would take the introduction of an entire new branch of mathematics in order to devise a proof for the rather simplistic looking equation. This document highlights the first major steps taken in proving the theorem, focusing on Kummer's proof for regular primes and the concepts that resulted. In particular Kummer's ideal numbers will be discussed as well as how they served as the precursors to ideals in ring theory.

Combinatorially Derived Properties Of Young Tableaux, 2014 College of William and Mary

#### Combinatorially Derived Properties Of Young Tableaux, James R. Janopaul-Naylor

*Undergraduate Honors Theses*

We examine properties of Young tableaux of shape λ and weight μ or of shape {λ_{(i)}}, a sequence of partitions. First we use combinatorial arguments to re- derive results about individual tableaux from Behrenstein and Zelevinskii regard- ing Kostka numbers and from Gates, Goldman, and Vinroot regarding when the weight μ on a tableau of shape λ is the unique weight with K_{λμ} = 1. Second we generalize these results to sequences of tableaux. Specifically we show under what conditions is K_{{λ(i)}μ} = 1 for a sequence of partitions {λ(i)} and weight μ and when is ...

The Combinatorialization Of Linear Recurrences, 2014 Harvey Mudd College

#### The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn

*Jennifer J. Quinn*

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

Mod 2 Homology For Gl(4) And Galois Representations, 2014 University of Massachusetts - Amherst

#### Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell

*Paul Gunnells*

We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ with F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a ...

A Frobenius Question Related To Actions On Curves In Characteristic P, 2014 Gettysburg College

#### A Frobenius Question Related To Actions On Curves In Characteristic P, Darren B. Glass

*Math Faculty Publications*

We consider which integers *g* can occur as the genus and of a curve defined over a field of characteristic *p* which admits an automorphism of degree *pq*, where *p* and *q* are distinct primes. This investigation leads us to consider a certain family of three-dimensional Frobenius problems and prove explicit formulas giving their solution in many cases.

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, 2014 Illinois Wesleyan University

#### Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

*Scholarship*

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with prime factors for every between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes with the property that divides a highly composite .

There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, 2014 Harvey Mudd College

#### There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich

*HMC Senior Theses*

*L*-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

Finding Zeros Of Rational Quadratic Forms, 2014 Claremont McKenna College

#### Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy

*CMC Senior Theses*

In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

Computing Local Constants For Cm Elliptic Curves, 2014 College of Saint Benedict/Saint John's University

#### Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

*Mathematics Faculty Publications*

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

Computing Boundary Extensions Of Conformal Maps, 2014 Iowa State University

#### Computing Boundary Extensions Of Conformal Maps, Timothy H. Mcnicholl

*Mathematics Publications*

We show that a computable and conformal map of the unit disk onto a bounded domain D has a computable boundary extension if D has a computable boundary connectivity function.

Explorations Of The Collatz Conjecture (Mod M), 2014 Georgia Southern University

#### Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr

*University Honors Program Theses*

The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and ...

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, 2013 Illinois Wesleyan University

#### Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

*Andrew Shallue*

The Kronecker-Weber Theorem: An Exposition, 2013 Lawrence University

#### The Kronecker-Weber Theorem: An Exposition, Amber Verser

*Lawrence University Honors Projects*

This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in *The American Mathematical Monthly* in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number ...

Quotients Of Gaussian Primes, 2013 Pomona College

#### Quotients Of Gaussian Primes, Stephan Ramon Garcia

*Pomona Faculty Publications and Research*

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"

Wilson Theorem, 2013 Selected Works

Rock Art Tallies: Mathematics On Stone In Western North America, 2013 Millikin University

#### Rock Art Tallies: Mathematics On Stone In Western North America, James V. Rauff

*Journal of Humanistic Mathematics*

Western North America abounds with rock art sites. From Alberta to New Mexico and from Minnesota to California one can find the enigmatic rock paintings and rock carvings left by the pre-Columbian inhabitants. The images left behind on the rocks of the American plains and deserts are those of humanoids and animals, arrows and spears, and a variety of geometric shapes and abstract designs. Also included, in great numbers, are sequences of repeated shapes and marks that scholars have termed "tallies." The tallies are presumed to be an ancient accounting of something or some things. This article examines rock art ...

Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, 2013 Universitat Duisburg-Essen, Germany

#### Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann

*Peter Loth*

Warfield modules are direct summands of simply presented Zp - modules, or, alternatively, are Zp - modules possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by theor Ilm-Kaplansky and Warfield invariants. Taking a model theoretic point of view and using infinitary languages we give here a complete theoretic characterization of a large class of Zp - modules having a nice decomposition basis. As a corollary, we obtain the classical classification of countable Warfield modules. This generalizes results by Barwise and Eklof.

Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, 2013 Scripps College

#### Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, Tracy Layden

*Scripps Senior Theses*

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Slicing A Puzzle And Finding The Hidden Pieces, 2013 Olivet Nazarene University

#### Slicing A Puzzle And Finding The Hidden Pieces, Martha Arntson

*Honors Program Projects*

The research conducted was to investigate the potential connections between group theory and a puzzle set up by color cubes. The goal of the research was to investigate different sized puzzles and discover any relationships between solutions of the same sized puzzles. In this research, first, there was an extensive look into the background of Abstract Algebra and group theory, which is briefly covered in the introduction. Then, each puzzle of various sizes was explored to find all possible color combinations of the solutions. Specifically, the 2x2x2, 3x3x3, and 4x4x4 puzzles were examined to find that the 2x2x2 has 24 ...