Integral Generalized Binomial Coefficients Of Multiplicative Functions, 2015 University of Puget Sound

#### Integral Generalized Binomial Coefficients Of Multiplicative Functions, Imanuel Chen

*Summer Research*

The binomial coefficients are interestingly always integral. However, when you generalize the binomial coefficients to any class of function, this is not always the case. Multiplicative functions satisfy the properties: f(ab) = f(a)f(b) when a and b are relatively prime, and f(1) = 1. Tom Edgar of Pacific Lutheran University and Michael Spivey of the University of Puget Sound developed a Corollary that determines which values of n and m will always have integral generalized binomial coefficients for all multiplicative functions. The purpose of this research was to determine as many patterns within this corollary as possible ...

Basis Criteria For N-Cycle Integer Splines, 2015 Bard College

#### Basis Criteria For N-Cycle Integer Splines, Ester Gjoni

*Senior Projects Spring 2015*

In this project we work with integer splines on graphs with positive integer edge labels. We focus on graphs that are n-cycles for some natural number n. We find an explicit condition for when a set of splines can form a module basis for n-cycle splines. In general, a set of splines forms a Z-module basis if and only if their determinant is equal to the product of the edge labels divided by the greatest common divisor of those edge labels.

Drawing Numbers And Listening To Patterns, 2015 Georgia Southern University

#### Drawing Numbers And Listening To Patterns, Loren Zo Haynes

*University Honors Program 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 ...

Irreducibility And Galois Properties Of Lifts Of Supersingular Polynomials, 2015 Bard College

#### Irreducibility And Galois Properties Of Lifts Of Supersingular Polynomials, Rylan Jacob Gajek-Leonard

*Senior Projects Spring 2015*

It has recently been shown that a rational specialization of Jacobi polynomials, when reduced modulo a prime number p, has roots which coincide with the supersingular j- invariants of elliptic curves in characteristic p. These supersingular lifts are conjectured to be irreducible with maximal Galois groups. Using the theory of p-adic Newton Polygons, we provide a new infinite class of irreducibility and, assuming a conjecture of Hardy and Littlewood, give strong evidence for their Galois groups being as large as possible.

Rational Tilings Of The Unit Square, 2015 Bard College

#### Rational Tilings Of The Unit Square, Galen Dorpalen-Barry

*Senior Projects Spring 2015*

A rational n-tiling of the unit square is a collection of n triangles with rational side length whose union is the unit square and whose intersections are at most their boundary edges. It is known that there are no rational 2-tilings or 3-tilings of the unit square, and that there are rational 4- and 5-tilings. The nature of those tilings is the subject of current research. In this project we give a combinatorial basis for rational n-tilings and explore rational 6-tilings of the unit square.

On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, 2015 Claremont McKenna College

#### On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh

*CMC Senior Theses*

This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.

Elliptic Curves And The Congruent Number Problem, 2015 Claremont McKenna College

#### Elliptic Curves And The Congruent Number Problem, Jonathan Star

*CMC Senior Theses*

In this paper we explain the congruent number problem and its connection to elliptic curves. We begin with a brief history of the problem and some early attempts to understand congruent numbers. We then introduce elliptic curves and many of their basic properties, as well as explain a few key theorems in the study of elliptic curves. Following this, we prove that determining whether or not a number *n* is congruent is equivalent to determining whether or not the algebraic rank of a corresponding elliptic curve E_{n} is 0. We then introduce *L*-functions and explain the Birch and ...

A Cryptographic Attack: Finding The Discrete Logarithm On Elliptic Curves Of Trace One, 2015 Scripps College

#### A Cryptographic Attack: Finding The Discrete Logarithm On Elliptic Curves Of Trace One, Tatiana Bradley

*Scripps Senior Theses*

The crux of elliptic curve cryptography, a popular mechanism for securing data, is an asymmetric problem. The elliptic curve discrete logarithm problem, as it is called, is hoped to be generally hard in one direction but not the other, and it is this asymmetry that makes it secure.

This paper describes the mathematics (and some of the computer science) necessary to understand and compute an attack on the elliptic curve discrete logarithm problem that works in a special case. The algorithm, proposed by Nigel Smart, renders the elliptic curve discrete logarithm problem easy in both directions for elliptic curves of ...

Obstruction Criteria For Modular Deformation Problems, 2015 University of Massachusetts - Amherst

#### Obstruction Criteria For Modular Deformation Problems, Jeffrey Hatley Jr

*Doctoral Dissertations*

For a cuspidal newform f of weight k at least 3 and a prime p of the associated number field K_{f}, the deformation problem for its associated mod p Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for f of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the p-adic deformation problem for f is unobstructed, then f is not congruent mod p to a newform of lower level.

Cohomology Of Absolute Galois Groups, 2014 The University of Western Ontario

#### Cohomology Of Absolute Galois Groups, Claudio Quadrelli

*Electronic Thesis and Dissertation Repository*

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-p case, i.e., one would like to know which pro-p groups occur as maximal pro-p Galois groups, i.e., maximal pro-p quotients of absolute Galois groups. Indeed, pro-p groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group.

We define a new class of pro-p groups ...

Deconstructing The Welch Equation Using P-Adic Methods, 2014 Rose-Hulman Institute of Technology

#### Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh

*Rose-Hulman Undergraduate Research Publications*

The Welch map x -> g^{x-1+c} is similar to the discrete exponential map x -> g^{x}, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: g^{x-1+c} = x (mod p^{e}) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo p^{e}, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p^{e}. These methods include ...

Deconstructing The Welch Equation Using P-Adic Methods, 2014 Rose-Hulman Institute of Technology

#### Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh

*Mathematical Sciences Technical Reports (MSTR)*

The Welch map x -> g^{x-1+c} is similar to the discrete exponential map x -> g^{x}, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: g^{x-1+c} = x (mod p^{e}) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo p^{e}, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p^{e}. These methods include ...

Prime Decomposition In Iterated Towers And Discriminant Formulae, 2014 University of Massachusetts - Amherst

#### Prime Decomposition In Iterated Towers And Discriminant Formulae, Thomas Alden Gassert

*Doctoral Dissertations*

We explore certain arithmetic properties of iterated extensions. Namely, we compute the index associated to certain families of iterated polynomials and determine the decomposition of prime ideals in others.

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 ...

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 .

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.