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Full-Text Articles in Number Theory

Determinant Formulas Of Some Hessenberg Matrices With Jacobsthal Entries, Taras Goy, Mark Shattuck Jun 2021

Determinant Formulas Of Some Hessenberg Matrices With Jacobsthal Entries, Taras Goy, Mark Shattuck

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we evaluate determinants of several families of Hessenberg matrices having various subsequences of the Jacobsthal sequence as their nonzero entries. These identities may be written equivalently as formulas for certain linearly recurrent sequences expressed in terms of sums of products of Jacobsthal numbers with multinomial coefficients. Among the sequences that arise in this way include the Mersenne, Lucas and Jacobsthal-Lucas numbers as well as the squares of the Jacobsthal and Mersenne sequences. These results are extended to Hessenberg determinants involving sequences that are derived from two general families of linear second-order recurrences. Finally, combinatorial proofs are provided ...


Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell May 2021

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, Stephen Wang May 2021

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 ...


2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen May 2021

2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen

Honors Theses

The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other sequences related ...


A Weighted Version Of Erdős-Kac Theorem, Unique Subedi May 2021

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 ...


On Elliptic Curves, Montana S. Miller May 2021

On Elliptic Curves, Montana S. Miller

MSU Graduate Theses

An elliptic curve over the rational numbers is given by the equation y2 = x3+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.


A History And Translation Of Lagrange's "Sur Quelques Problèmes De L'Analyse De Diophante'', Christopher Goff, Michael Saclolo Feb 2021

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, Al Cuoco, E. Paul Goldenberg Jan 2021

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, Shalome Kurian Jan 2021

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, Nicasio M. Velez Jan 2021

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.


The Plus-Minus Davenport Constant Of Finite Abelian Groups, Darleen S. Perez-Lavin Jan 2021

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 log2 n ...


The Smallest Solution Of An Isotropic Quadratic Form, Deborah H. Blevins Jan 2021

The Smallest Solution Of An Isotropic Quadratic Form, Deborah H. Blevins

Theses and Dissertations--Mathematics

An isotropic quadratic form f(x1,...,xn) = ∑ ni=1nj=1 fijxixj defined on a Z- lattice has a smallest solution, where the size of the solution is measured using the infinity norm (∥ ∥), the l1 norm (∥ ∥1), or the Euclidean norm (∥ ∥2). Much work has been done to find the least upper bound and greatest lower bound on the smallest solution, beginning with Cassels in the mid-1950’s. Defining F := (f11,...,f1n,f21,...,f2n,...,fn1,...,fnn), upper bound results have the ...


On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang Jan 2021

On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang

CMC Senior Theses

In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative coordinates in L ...


An Introduction To Number Theory, J. J. P. Veerman Jan 2021

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).


Tiling Representations Of Zeckendorf Decompositions, John Lentfer Jan 2021

Tiling Representations Of Zeckendorf Decompositions, John Lentfer

HMC Senior Theses

Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f1 = 1 and f2 = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf ...


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 Dec 2020

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, Mohammad Shatnawi Dec 2020

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, Obiamaka L. Agu Dec 2020

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, Christian Carley Nov 2020

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, Morgan Bounds Nov 2020

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, Anthony Teeter, Margo Bergman Nov 2020

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, Florentin Smarandache, Nivetha Martin, Priya R. Oct 2020

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, Darren B. Glass, Joshua R. Wagner Aug 2020

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, Daniel A. Goldston, Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, Cem Y. Yıldırım Jul 2020

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, Mary K. Flagg Jul 2020

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

Number Theory

No abstract provided.


Harmony Amid Chaos, Drew Schaffner Jul 2020

Harmony Amid Chaos, Drew Schaffner

Pence-Boyce STEM Student Scholarship

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as ...


Multiparty Non-Interactive Key Exchange And More From Isogenies On Elliptic Curves, Dan Boneh, Darren B. Glass, Daniel Krashen, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, Mark Zhandry Jun 2020

Multiparty Non-Interactive Key Exchange And More From Isogenies On Elliptic Curves, Dan Boneh, Darren B. Glass, Daniel Krashen, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, Mark Zhandry

Math Faculty Publications

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety.

Our framework builds a cryptographic invariant ...


Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek Jun 2020

Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek

Dissertations, Theses, and Capstone Projects

A result by Fueter-Pólya states that the only quadratic polynomials that bijectively map the integral lattice points of the first quadrant onto the non-negative integers are the two Cantor polynomials. We study the more general case of bijective mappings of quadratic polynomials from the lattice points of sectors defined as the convex hull of two rays emanating from the origin, one of which falls along the x-axis, the other being defined by some vector. The sector is considered rational or irrational according to whether this vector can be written with rational coordinates or not. We show that the existence of ...


Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan Jun 2020

Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan

Dissertations, Theses, and Capstone Projects

Harris and Kudla (2004) proved a conjecture of Jacquet, that the central value of a triple product L-function does not vanish if and only if there exists a quaternion algebra over which a period integral of three corresponding automorphic forms does not vanish. Moreover, Gross and Kudla (1992) established an explicit identity relating central L-values and period integrals (which are finite sums in their case), when the cusp forms are of prime levels and weight 2. Böcherer, Schulze-Pillot (1996) and Watson (2002) generalized this identity to more general levels and weights, and Ichino (2008) proved an adelic period formula which ...


The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes May 2020

The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes

Student Theses

For a pair of quadratic integers n and m chosen randomly, uniformly, and independently from the set of quadratic integers of norm x or less, we calculate the probability that the greatest common divisor of (n,m) is k. We also calculate the expected norm of the greatest common divisor (n,m) as x tends to infinity, with explicit error terms. We determine the probability and expected norm of the greatest common divisor for quadratic integer rings that are unique factorization domains. We also outline a method to determine the probability and expected norm of the greatest common divisor of ...