Greatest Common Divisor: Algorithm And Proof, 2019 University of St. Thomas - Houston

#### Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg

*Number Theory*

No abstract provided.

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

*Rose-Hulman Undergraduate Mathematics Journal*

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 ...

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

Experience Of A Noyce-Student Learning Assistant In An Inquiry-Based Learning Class, 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 ...

Sums Involving The Number Of Distinct Prime Factors Function, 2018 University of Maryland, College Park

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

*Rose-Hulman Undergraduate Mathematics Journal*

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

*Rose-Hulman Undergraduate Mathematics Journal*

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.

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 ...

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 ...

The Distribution Of Totally Positive Integers In Totally Real Number Fields, 2018 The Graduate Center, City University of New York

#### The Distribution Of Totally Positive Integers In Totally Real Number Fields, Tianyi Mao

*All Dissertations, Theses, and Capstone Projects*

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the number field is quadratic, Beck also proved a mean value result using the continued fraction expansions of quadratic irrationals. We generalize Beck’s result to higher moments. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the ...

Secure Multiparty Protocol For Differentially-Private Data Release, 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 ...

Quantum Attacks On Modern Cryptography And Post-Quantum Cryptosystems, 2018 Liberty University

#### Quantum Attacks On Modern Cryptography And Post-Quantum Cryptosystems, Zachary Marron

*Senior Honors Theses*

Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial ...

Pgl2(FL) Number Fields With Rational Companion Forms, 2018 University of Minnesota - Morris

#### Pgl2(FL) Number Fields With Rational Companion Forms, David P. Roberts

*Mathematics Publications*

We give a list of PGL_{2}(F_{l}) number fields for ℓ ≥ 11 which have rational companion forms. Our list has fifty-three fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.

Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, 2018 Sacred Heart University

#### Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, Sarah Riccio

*Mathematics Undergraduate Publications*

In this paper, three topics in number theory will be explored: Niven Numbers, the Factorial Triangle, and Erdos's Conjecture . For each of these topics, the goal is for us to find patterns within the numbers which help us determine all possible values in each category. We will look at two digit Niven Numbers and the set that they belong to, the alternating summation of the rows of the Factorial Triangle, and the unit fractions whose sum is the basis of Erdos' Conjecture.

Monomial Progenitors And Related Topics, 2018 California State University - San Bernardino

#### Monomial Progenitors And Related Topics, Madai Obaid Alnominy

*Electronic Theses, Projects, and Dissertations*

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M_{11}, HS × D_{5}, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L_{2}(149) as homomorphic images of the monomial progenitors 11*^{4} :_{m} (5 :4), 5*^{6 } :_{m} S_{5} and 149*^{2 } :_{m } D_{37}. We have also discovered 2^{4} : S_{3} × C_{2}, 2 ...

On The Density Of The Odd Values Of The Partition Function, 2018 Michigan Technological University

#### On The Density Of The Odd Values Of The Partition Function, Samuel Judge

*Dissertations, Master's Theses and Master's Reports*

The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities ...

Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, 2018 Murray State University

#### Distributive Lattice Models Of The Type C One-Rowed Weyl Group Symmetric Functions, William Atkins

*Murray State Theses and Dissertations*

We present two families of diamond-colored distributive lattices – one known and one new – that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ﬁlling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with ...

Equidimensional Adic Eigenvarieties For Groups With Discrete Series, 2018 Illinois Mathematics and Science Academy

#### Equidimensional Adic Eigenvarieties For Groups With Discrete Series, Daniel Robert Gulotta '03

*Doctoral Dissertations*

We extend Urban's construction of eigenvarieties for reductive groups *G* such that *G*(R) has discrete series to include characteristic *p* points at the boundary of weight space. In order to perform this construction, we define a notion of "locally analytic" functions and distributions on a locally Q_{p}-analytic manifold taking values in a complete Tate Z_{p}-algebra in which *p* is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on *p*-adic Lie groups given by Johansson and Newton.

An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, 2018 The University of Akron

#### An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, Jacob Liddy

*Williams Honors College, Honors Research Projects*

An abundant number is said to be primitive if none of its proper divisors are abundant. Dickson proved that for an arbitrary positive integer d there exists only finitely many odd primitive abundant numbers having exactly d prime divisors. In this paper we describe a fast algorithm that finds all primitive odd numbers with d unique prime divisors. We use this algorithm to find all the number of odd primitive abundant numbers with 6 unique Divisors. We use this algorithm to prove that an odd weird number must have at least 6 prime divisors.