# Number Theory Commons™

## All Articles in Number Theory

429 full-text articles. Page 5 of 17.

New Experimental Investigations For The 3x+1 Problem: The Binary Projection Of The Collatz Map, 2019 University of California, Davis

#### New Experimental Investigations For The 3x+1 Problem: The Binary Projection Of The Collatz Map, Benjamin Bairrington, Aaron Okano

The 3x + 1 Problem, or the Collatz Conjecture, was originally developed in the early 1930's. It has remained unsolved for over eighty years. Throughout its history, traditional methods of mathematical problem solving have only succeeded in proving heuristic properties of the mapping. Because the problem has proven to be so difficult to solve, many think it might be undecidable. In this paper we brie y follow the history of the 3x + 1 problem from its creation in the 1930's to the modern day. Its history is tied into the development of the Cosper Algorithm, which maps binary sequences ...

Enhanced Koszulity In Galois Cohomology, 2019 The University of Western Ontario

#### Enhanced Koszulity In Galois Cohomology, Marina Palaisti

##### Electronic Thesis and Dissertation Repository

Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois ...

2019 University of Nebraska - Lincoln

#### Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke

##### Honors Theses, University of Nebraska-Lincoln

In this thesis, Pascal's Triangle modulo n will be explored for n prime and n a prime power. Using the results from the case when n is prime, a novel proof of Lucas' Theorem is given. Additionally, using both the results from the exploration of Pascal's Triangle here, as well as previous results, an efficient algorithm for computation of binomial coefficients modulo n (a choose b mod n) is described, and its time complexity is analyzed and compared to naive methods. In particular, the efficient algorithm runs in O(n log(a)) time (as opposed to the naive ...

2019 University of Nebraska at Omaha

#### Experience Of A Noyce-Student Learning Assistant In An Inquiry-Based Learning Class, Melissa Riley

##### Student Research and Creative Activity Fair

This presentation refers to an undergraduate course called introduction to abstract mathematics at the University of Nebraska at Omaha. During the academic year 2017-2018, undergraduate, mathematics student Melissa Riley was a Noyce-student learning assistant for the Inquiry Based Learning (IBL) section of the course. She assisted the faculty-in-charge with all aspects of the course. These included: materials preparation, class organization, teamwork, class leading, presentations, and tutoring. This presentation shall address some examples of how the IBL approach can be used in this type of class including: the structure of the course, the activities and tasks performed by the students, learning ...

A Short Remark On Gödel Incompleteness Theorem And Its Self-Referential Paradox From Neutrosophic Logic Perspective, 2019 University of New Mexico

#### A Short Remark On Gödel Incompleteness Theorem And Its Self-Referential Paradox From Neutrosophic Logic Perspective, Florentin Smarandache, Victor Christianto

##### Mathematics and Statistics Faculty and Staff Publications

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective ...

2019 University of New Mexico

#### Special Issue: New Types Of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/ Off-Set, Neutrosophic Refined Set, And Their Extension To Plithogenic Set/Logic/ Probability, With Applications, Florentin Smarandache

##### Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Special Issue: Algebraic Structures Of Neutrosophic Triplets, Neutrosophic Duplets, Or Neutrosophic Multisets, Vol. Ii, 2019 University of New Mexico

#### Special Issue: Algebraic Structures Of Neutrosophic Triplets, Neutrosophic Duplets, Or Neutrosophic Multisets, Vol. Ii, Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali

##### Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Factorization Lengths In Numerical Monoids, 2019 Bard College

#### Factorization Lengths In Numerical Monoids, Maya Samantha Schwartz

##### Senior Projects Spring 2019

A numerical monoid M generated by the natural numbers n_1, ..., n_k is a subset of {0, 1, 2, ...} whose elements are non-negative linear combinations of the generators n_1, ..., n_k. The set of factorizations of an element in M is the set of all the different ways to write that element as a linear combination of the generators. The length of a factorization of an element is the sum of the coefficients of that factorization. Since an element in a monoid can be written in different ways in terms of the generators, its set of factorization lengths may contain more than ...

2019 Bard College

#### The Conditional Probability That An Elliptic Curve Has A Rational Subgroup Of Order 5 Or 7, Meagan Kenney

##### Senior Projects Spring 2019

Let E be an elliptic curve over the rationals. There are two different ways in which the set of rational points on E can be said to be divisible by a prime p. We will call one of these types of divisibility local and the other global. Global divisibility will imply local divisibility; however, the converse is not guaranteed. In this project we focus on the cases where p=5 and p=7 to determine the probability that E has global divisibility by p, given that E has local divisibility by p.

Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, 2019 University of Kentucky

#### Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, Karthik Chandrasekhar

##### Theses and Dissertations--Mathematics

A tournament graph G is a vertex set V of size n, together with a directed edge set EV × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, jV and (i, i) ∉ E for all iV. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the (n − 1)-dimensional simplex on V is given an orientation. In this dissertation we bound the number of compatible k-simplices ...

Hidden Symmetries In Classical Mechanics And Related Number Theory Dynamical System, 2019 Eastern Illinois University

#### Hidden Symmetries In Classical Mechanics And Related Number Theory Dynamical System, Mohsin Md Abdul Karim

##### Masters Theses

Classical Mechanics consists of three parts: Newtonian, Lagrangian and Hamiltonian Mechanics, where each part is a special extension of the previous part. Each part has explicit symmetries (the explicit Laws of Motion), which, in turn, generate implicit or hidden symmetries (like the Law of Conservation of Energy, etc). In this Master's Thesis, different types of hidden symmetries are considered; they are reflected in the Noether Theorem and the Poincare Recurrence Theorem applied to Lagrangian and Hamiltonian Systems respectively.

The Poincare Recurrence Theorem is also applicable to some number theory problems, which can be considered as dynamical systems. In this ...

Primes In Arithmetical Progression, 2019 Colby College

#### Primes In Arithmetical Progression, Edward C. Wessel

##### Honors Theses

This thesis will tackle Dirichlet’s Theorem on Primes in Arithmetical Progressions. The majority of information that follows below will stem from Tom M. Apostol’s Introduction to Analytical Number Theory. This is the main source of all definitions, theorems, and method. However, I would like to assure the reader that prior knowledge of neither the text nor analytical number theory in general is needed to understand the result. A rough background in Abstract Algebra and a moderate grasp on Complex and Real Analysis are more than sufficient. In fact, my project’s intent is to introduce Dirichlet’s ideas ...

Triangular Cubic Hesitant Fuzzy Einstein Hybrid Weighted Averaging Operator And Its Application To Decision Making, 2018 University of New Mexico

#### Triangular Cubic Hesitant Fuzzy Einstein Hybrid Weighted Averaging Operator And Its Application To Decision Making, Florentin Smarandache, Aliya Fahmi, Fazli Amin, Madad Khan, Nasruddin Hassan

##### Mathematics and Statistics Faculty and Staff Publications

In this paper, triangular cubic hesitant fuzzy Einstein weighted averaging (TCHFEWA) operator, triangular cubic hesitant fuzzy Einstein ordered weighted averaging (TCHFEOWA) operator and triangular cubic hesitant fuzzy Einstein hybrid weighted averaging (TCHFEHWA) operator are proposed. An approach to multiple attribute group decision making with linguistic information is developed based on the TCHFEWA and the TCHFEHWA operators. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the existing aggregation operators. Finally, a numerical example is provided to demonstrate the application of the established approach

2018 University of Maryland, College Park

#### Sums Involving The Number Of Distinct Prime Factors Function, Tanay Wakhare

We find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria for these series. The approach of this paper is to use the theory of symmetric functions to derive identities for the elementary symmetric functions, then apply these identities to arbitrary primes and values of multiplicative functions evaluated at primes. This allows us to reinterpret sums over symmetric polynomials as divisor sums and sums over the natural numbers.

On Orders Of Elliptic Curves Over Finite Fields, 2018 Columbia University

#### On Orders Of Elliptic Curves Over Finite Fields, Yujin H. Kim, Jackson Bahr, Eric Neyman, Gregory Taylor

In this work, we completely characterize by $j$-invariant the number of orders of elliptic curves over all finite fields $F_{p^r}$ using combinatorial arguments and elementary number theory. Whenever possible, we state and prove exactly which orders can be taken on.

The Origin Of The Prime Number Theorem, 2018 Central Washington University

#### The Origin Of The Prime Number Theorem, Dominic Klyve

##### Number Theory

No abstract provided.

Character Sums Of Lee And Weintraub, 2018 CUNY New York City College of Technology

#### Character Sums Of Lee And Weintraub, Brad Isaacson

##### Publications and Research

We prove two conjectures of Lee and Weintraub and one conjecture of Ibukiyama and Kaneko about character sums arising as fixed point contributions in the Atiyah–Singer holomorphic Lefshetz formula applied to finite group actions on the space of certain Siegel cusp forms.

Modern Cryptography, 2018 California State University - San Bernardino

#### Modern Cryptography, Samuel Lopez

##### Electronic Theses, Projects, and Dissertations

We live in an age where we willingly provide our social security number, credit card information, home address and countless other sensitive information over the Internet. Whether you are buying a phone case from Amazon, sending in an on-line job application, or logging into your on-line bank account, you trust that the sensitive data you enter is secure. As our technology and computing power become more sophisticated, so do the tools used by potential hackers to our information. In this paper, the underlying mathematics within ciphers will be looked at to understand the security of modern ciphers.

An extremely important ...

2018 Boise State University

#### Secure Multiparty Protocol For Differentially-Private Data Release, Anthony Harris

##### Boise State University Theses and Dissertations

In the era where big data is the new norm, a higher emphasis has been placed on models which guarantees the release and exchange of data. The need for privacy-preserving data arose as more sophisticated data-mining techniques led to breaches of sensitive information. In this thesis, we present a secure multiparty protocol for the purpose of integrating multiple datasets simultaneously such that the contents of each dataset is not revealed to any of the data owners, and the contents of the integrated data do not compromise individual’s privacy. We utilize privacy by simulation to prove that the protocol is ...

Vector Partitions, 2018 East Tennessee State University

#### Vector Partitions, Jennifer French

##### Electronic Theses and Dissertations

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The ...