Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), 2021 Jacksonville State University

#### Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell

*Theses*

This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued ...

Zeta Function Regularization And Its Relationship To Number Theory, 2021 East Tennessee State University

#### Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

*Electronic Theses and Dissertations*

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to ...

A Weighted Version Of Erdős-Kac Theorem, 2021 University of Mississippi

#### A Weighted Version Of Erdős-Kac Theorem, Unique Subedi

*Honors Theses*

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. A celebrated result of Erd{\H o}s and Kac states that $\omega(n)$ as a Gaussian distribution. In this thesis, we establish a weighted version of Erd{\H o}s-Kac Theorem. Specifically, we show that the Gaussian limiting distribution is preserved, but shifted, when $\omega(n)$ is weighted by the $k-$fold divisor function $\tau_k(n)$. We establish this result by computing all positive integral moments of $\omega(n)$ weighted by $\tau_k(n)$.

We also provide a proof of the classical identity of $\zeta ...

A History And Translation Of Lagrange's "Sur Quelques Problèmes De L'Analyse De Diophante'', 2021 University of the Pacific

#### A History And Translation Of Lagrange's "Sur Quelques Problèmes De L'Analyse De Diophante'', Christopher Goff, Michael Saclolo

*Euleriana*

Among Lagrange's many achievements in number theory is a solution to the problem posed and solved by Fermat of finding a right triangle whose legs sum to a perfect square and whose hypotenuse is also a square. This article chronicles various appearances of the problem, including multiple solutions by Euler, all of which inadequately address completeness and minimality of solutions. Finally, we summarize and translate Lagrange's paper in which he solves the problem completely, thus successfully proving the minimality of Fermat's original solution.

Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head, 2021 Education Development Center

#### Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head, Al Cuoco, E. Paul Goldenberg

*Journal of Humanistic Mathematics*

How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and ...

A Case Study On Hooley's Conditional Proof Of Artin's Primitive Root Conjecture, 2021 University of Warwick, Coventry

#### A Case Study On Hooley's Conditional Proof Of Artin's Primitive Root Conjecture, Shalome Kurian

*Rose-Hulman Undergraduate Mathematics Journal*

Artin’s Primitive Root Conjecture represents one of many famous problems in elementary number theory that has resisted complete solution thus far. Significant progress was made in 1967, when Christopher Hooley published a conditional proof of the conjecture under the assumption of a certain case of the Generalised Riemann Hypothesis. In this survey we present a description of the conjecture and the underlying algebraic theory, and provide a detailed account of Hooley’s proof which is intended to be accessible to those with only undergraduate level knowledge. We also discuss a result concerning the qx+1 problem, whose proof requires ...

Mathematical Magic: A Study Of Number Puzzles, 2021 Maryville College

#### Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez

*Rose-Hulman Undergraduate Mathematics Journal*

Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.

An Introduction To Number Theory, 2021 Portland State University

#### An Introduction To Number Theory, J. J. P. Veerman

*PDXOpen: Open Educational Resources*

These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed.

Chapters 1-6 represent approximately 1 trimester of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. The current content represents a course the author taught in Fall 2020.

It is a work in progress. If you have questions or comments, please contact Peter Veerman (veerman@pdx.edu).

The Plus-Minus Davenport Constant Of Finite Abelian Groups, 2021 University of Kentucky

#### The Plus-Minus Davenport Constant Of Finite Abelian Groups, Darleen S. Perez-Lavin

*Theses and Dissertations--Mathematics*

Let G be a finite abelian group, written additively. The Davenport constant, *D(G)*, is the smallest positive number *s* such that any subset of the group *G*, with cardinality at least *s*, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, *D*_{±}(*G*). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of *G* is less than or equal to 100, then the *D*_{±}(*G*) is the floor of log_{2} *n ...*

On Elliptic Curves, 2021 Missouri State University

#### On Elliptic Curves, Montana S. Miller

*MSU Graduate Theses*

An elliptic curve over the rational numbers is given by the equation *y ^{2} = x^{3}+Ax+B*. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.

On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes, 2020 University of Gothenburg

#### On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes, J. Brandes, Scott T. Parsell, C. Poulias, G. Shakan, R. C. Vaughn

*Mathematics Faculty Publications*

We obtain asymptotics for sums of the form

Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),

involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has

sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon,

and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.

On The Local Theory Of Profinite Groups, 2020 Western Michigan University

#### On The Local Theory Of Profinite Groups, Mohammad Shatnawi

*Dissertations*

Let *G* be a finite group, and *H* be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of *H.* The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group *G* into *H/H ^{'}* an abelian group where

*H*is a subgroup of

*G*and

*H'*is the derived group of

*H*. One important first application is Burnside’s normal

*p*-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it ...

Sum Of Cubes Of The First N Integers, 2020 California State University, San Bernardino

#### Sum Of Cubes Of The First N Integers, Obiamaka L. Agu

*Electronic Theses, Projects, and Dissertations*

In Calculus we learned that Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at ...

The Name Tag Problem, 2020 Boise State University

#### The Name Tag Problem, Christian Carley

*Rose-Hulman Undergraduate Mathematics Journal*

The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is ...

New Theorems For The Digraphs Of Commutative Rings, 2020 Indiana Wesleyan University

#### New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds

*Rose-Hulman Undergraduate Mathematics Journal*

The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli *a* and *b* are used to predict cycles in the digraph of the ring with modulus *ab*. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.

Applying The Data: Predictive Analytics In Sport, 2020 University of Washington, Tacoma

#### Applying The Data: Predictive Analytics In Sport, Anthony Teeter, Margo Bergman

*Access*: Interdisciplinary Journal of Student Research and Scholarship*

The history of wagering predictions and their impact on wide reaching disciplines such as statistics and economics dates to at least the 1700’s, if not before. Predicting the outcomes of sports is a multibillion-dollar business that capitalizes on these tools but is in constant development with the addition of big data analytics methods. Sportsline.com, a popular website for fantasy sports leagues, provides odds predictions in multiple sports, produces proprietary computer models of both winning and losing teams, and provides specific point estimates. To test likely candidates for inclusion in these prediction algorithms, the authors developed a computer model ...

Decision Making On Teachers’ Adaptation To Cybergogy In Saturated Interval- Valued Refined Neutrosophic Overset /Underset /Offset Environment, 2020 University of New Mexico

#### Decision Making On Teachers’ Adaptation To Cybergogy In Saturated Interval- Valued Refined Neutrosophic Overset /Underset /Offset Environment, Florentin Smarandache, Nivetha Martin, Priya R.

*Mathematics and Statistics Faculty and Staff Publications*

Neutrosophic overset, neutrosophic underset and neutrosophic offset introduced by Smarandache are the special kinds of neutrosophic sets with values beyond the range [0,1] and these sets are pragmatic in nature as it represents the real life situations. This paper introduces the concept of saturated refined neutrosophic sets and extends the same to the special kinds of neutrosophic sets. The proposed concept is applied in decision making on Teacher’s adaptation to cybergogy. The decision making environment is characterized by different types of teachers, online teaching skills and various training methods. Fuzzy relation is used to match the most suitable ...

Arithmetical Structures On Paths With A Doubled Edge, 2020 Gettysburg College

#### Arithmetical Structures On Paths With A Doubled Edge, Darren B. Glass, Joshua R. Wagner

*Math Faculty Publications*

An arithmetical structure on a graph is given by a labeling of the vertices that satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical structures for graphs in these families.

Small Gaps Between Almost Primes, The Parity Problem, And Some Conjectures Of Erdős On Consecutive Integers Ii, 2020 San Jose State University

#### Small Gaps Between Almost Primes, The Parity Problem, And Some Conjectures Of Erdős On Consecutive Integers Ii, Daniel A. Goldston, Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, Cem Y. Yıldırım

*Faculty Publications*

We show that for any positive integer *n*, there is some fixed *A* such that *d(x)* = *d(x +n)* = *A* infinitely often where *d(x) *denotes the number of divisors of *x*. In fact, we establish the stronger result that both *x* and *x +n* have the same fixed exponent pattern for infinitely many *x*. Here the exponent pattern of an integer *x* > 1is the multiset of nonzero exponents which appear in the prime factorization of *x*.

Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, 2020 University of St. Thomas - Houston

#### Cross-Cultural Comparisons: The Art Of Computing The Greatest Common Divisor, Mary K. Flagg

*Number Theory*

No abstract provided.