A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, 2012 Pomona College

#### A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali

*Pomona Faculty Publications and Research*

No abstract provided.

On A Quantum Form Of The Binomial Coefficient, 2012 University of Tennessee Space Institute

#### On A Quantum Form Of The Binomial Coefficient, Eric J. Jacob

*Masters Theses*

A unique form of the quantum binomial coefficient (n choose k) for k = 2 and 3 is presented in this thesis. An interesting double summation formula with floor function bounds is used for k = 3. The equations both show the discrete nature of the quantum form as the binomial coefficient is partitioned into specific quantum integers. The proof of these equations has been shown as well. The equations show that a general form of the quantum binomial coefficient with k summations appears to be feasible. This will be investigated in future work.

Elliptic Curves Of High Rank, 2012 Macalester College

#### Elliptic Curves Of High Rank, Cecylia Bocovich

*Mathematics, Statistics, and Computer Science Honors Projects*

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y^{2} = x^{3} + ax + b, where a and b are coefficients that satisfy the property 4a^{3} + 27b^{2} = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic to Z^{r} ⊕ E ...

Circular Units Of Function Fields, 2012 Rhode Island College

#### Circular Units Of Function Fields, Frederick Harrop

*Frederick F Harrop*

A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.

Computing Local Constants For Cm Elliptic Curves, 2012 College of Saint Benedict/Saint John's University

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

*Sunil Chetty*

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.

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, 2012 Irvine Valley College

#### Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

*All HMC Faculty Publications and Research*

We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

An Interesting Opportunity: The Gilbreath Conjecture, 2012 Carroll College, Helena, MT

#### An Interesting Opportunity: The Gilbreath Conjecture, Kyle Sturgill-Simon

*Mathematics, Engineering and Computer Science Undergraduate Theses*

Unsolved mathematical conjectures are a rich source of more than just frustration. By looking at the Gilbreath conjecture (unproven since 1878), we explore the learning opportunities that these unsolved problems present. Through explanation of the workings of the conjecture and the previous attempts by mathematicians the reader can develop an understanding of the number theory involved and even get started on a project of their own. This paper provides context for the Gilbreath conjecture so that an entry level mathematician (or person of any study) can get acquainted with an important but relatively unknown portion of mathematics. It also summarizes ...

Prove It!, 2012 Pomona College

#### Prove It!, Kenny W. Moran

*Journal of Humanistic Mathematics*

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Ramanujan Sums As Supercharacters, 2012 Pomona College

#### Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali

*Pomona Faculty Publications and Research*

The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.

Introduction Aux Méthodes D’Intégrale De Chemin Et Applications, 2012 SelectedWorks

#### Introduction Aux Méthodes D’Intégrale De Chemin Et Applications, Nour-Eddiine Fahssi

*Nour-Eddine Fahssi*

These lecture notes are based on a master course given at University Hassan II - Agdal in spring 2012.

Computing Local Constants Of Cm Elliptic Curves, 2011 College of Saint Benedict/Saint John's University

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

*Sunil Chetty*

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.

Resolutions Of The Steinberg Module For Gl(N), 2011 University of Massachusetts - Amherst

#### Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell

*Paul Gunnells*

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute ...

Constructing Large Numbers With Cheap Computers, 2011 Illinois Wesleyan University

#### Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman

*Scholarship*

No abstract provided.

Constructing Large Numbers With Cheap Computers, 2011 Illinois Wesleyan University

#### Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman

*Andrew Shallue*

No abstract provided.

Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, 2011 Universitat Duisburg-Essen, Germany

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

*Mathematics Faculty Publications*

Warfield modules are direct summands of simply presented **Z**_{p }- modules, or, alternatively, are **Z**_{p }- 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** Z**_{p }- 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.

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.