On Cyclotomic Primality Tests, 2011 University of Tennessee, Knoxville

#### On Cyclotomic Primality Tests, Thomas Francis Boucher

*Masters Theses*

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra's primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat's \little theorem" that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test ...

Higher Derivatives Of The Hurwitz Zeta Function, 2011 Western Kentucky University

#### Higher Derivatives Of The Hurwitz Zeta Function, Jason Musser

*Masters Theses & Specialist Projects*

The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s ...

The Elliptic Curve Discrete Logarithm And Functional Graphs, 2011 University of Arkansas

#### The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans

*Mathematical Sciences Technical Reports (MSTR)*

The discrete logarithm problem, and its adaptation to elliptic curves, called the elliptic curve discrete logarithm problem (ECDLP) is an open problem in the field of number theory, and its applications to modern cryptographic algorithms are numerous. This paper focuses on a statistical analysis of a modification to the ECDLP, called the x-ECDLP, where one is only given the xcoordinate of a point, instead of the entire point. Focusing only on elliptic curves whose field of definition is smaller than the number of points, this paper attempts to find a statistical indication of underlying structure (or lack thereof) in the ...

The Combinatorialization Of Linear Recurrences, 2011 Harvey Mudd College

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

*All HMC Faculty Publications and Research*

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.

Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, 2011 University of Massachusetts - Amherst

#### Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell

*Paul Gunnells*

In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms.

Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, 2011 University of Massachusetts - Amherst

#### Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, A Ash, Pe Gunnells, M Mcconnell

*Paul Gunnells*

In a previous paper [3] we computed cohomology groups H5(..0(N),C), where ..0(N) is a certain congruence subgroup of SL(4,Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.

On The Galois Group Of Generalized Laguerre Polynomials, 2011 University of Massachusetts - Amherst

#### On The Galois Group Of Generalized Laguerre Polynomials, F Hajir

*Farshid Hajir*

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α∈ℚ-ℤ <0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)=∑ j=0 n n+α n-j(-x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n.

Modular Forms And Elliptic Curves Over The Field Of Fifth Roots Of Unity, 2011 University of Massachusetts - Amherst

#### Modular Forms And Elliptic Curves Over The Field Of Fifth Roots Of Unity, Pe Gunnells, F Hajir, Dan Yasaki

*Farshid Hajir*

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F

Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, 2011 SelectedWorks

#### Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly

*Subhajit Kumar Ganguly*

The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.

Statistical Properties Of A Convoluted Beta-Weibull Distribution, 2011 Marshall University

#### Statistical Properties Of A Convoluted Beta-Weibull Distribution, Jianan Sun

*Theses, Dissertations and Capstones*

A new class of distributions recently developed involves the logit of the beta distribution. Among this class of distributions are the beta-normal (Eugene et.al. (2002)); beta-Gumbel (Nadarajah and Kotz (2004)); beta-exponential (Nadarajah and Kotz (2006)); beta-Weibull (Famoye et al. (2005)); beta-Rayleigh (Akinsete and Lowe (2008)); beta-Laplace (Kozubowski and Nadarajah (2008)); and beta-Pareto (Akinsete et al. (2008)), among a few others. Many useful statistical properties arising from these distributions and their applications to real life data have been discussed in the literature. One approach by which a new statistical distribution is generated is by the transformation of random variables having ...

Semi-Direct Galois Covers Of The Affine Line, 2011 Merrimack College

#### Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson

*Mathematics Faculty Publications*

Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/`Z) b o Z/pZ where b is a positive integer and ` is a prime distinct from p. In this paper, we study Galois covers ψ : Z → P 1 k ramified only over ∞ with Galois group G. We find the minimal genus of a curve Z which admits a covering map of this form and we give an explicit formula for this genus in terms of ` and p. The minimal genus occurs when b equals the order a of ...

Infinitesimal Time Scale Calculus, 2011 Marshall University

#### Infinitesimal Time Scale Calculus, Tom Cuchta

*Theses, Dissertations and Capstones*

Calculus has historically been fragmented into multiple distinct theories such as differential calculus, difference calculus, quantum calculus, and many others. These theories are all about the concept of what it means to "change", but in various contexts. Time scales calculus (introduced by Stefan Hilger in 1988) is a synthesis and extension of all the various calculi into a single theory. Calculus was originally approached with "infinitely small numbers" which fell out of use because the use of these numbers could not be justified. In 1960, Abraham Robinson introduced hyperreal numbers, a justification for their use, and therefore the original approach ...

Elliptic Curves: Minimally Spanning Prime Fields And Supersingularity, 2011 Bard College

#### Elliptic Curves: Minimally Spanning Prime Fields And Supersingularity, Travis Mcgrath

*Senior Projects Spring 2011*

Elliptic curves are cubic curves that have been studied throughout history. From Diophantus of Alexandria to modern-day cryptography, Elliptic Curves have been a central focus of mathematics. This project explores certain geometric properties of elliptic curves defined over finite fields.

Fix a finite field. This project starts by demonstrating that given enough elliptic curves, their union will contain every point in the affine plane. We then find the fewest curves possible such that their union still contains all these points. Using some of the tools discussed in solving this problem, we then explore what can be said about the number ...

Fractions Of Numerical Semigroups, 2010 University of Tennessee - Knoxville

#### Fractions Of Numerical Semigroups, Harold Justin Smith

*Doctoral Dissertations*

Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown.

Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is ...

On The Irreducibility Of The Cauchy-Mirimanoff Polynomials, 2010 University of Tennessee - Knoxville

#### On The Irreducibility Of The Cauchy-Mirimanoff Polynomials, Brian C. Irick

*Doctoral Dissertations*

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.

This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of ...

The Fibonacci Sequence, 2010 Parkland College

#### The Fibonacci Sequence, Arik Avagyan

*A with Honors Projects*

A review was made of the Fibonacci sequence, its characteristics and applications.

Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, 2010 Claremont McKenna College

#### Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger

*CMC Senior Theses*

Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.