Number Theory Commons^™

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

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

On A Generalization Of A Theorem Of Ibukiyama, Brad Isaacson

Publications and Research

We generalize a theorem of Ibukiyama and express periodic generalized Bernoulli functions by generalized Bernoulli numbers. As a corollary, we obtain formulas expressing these character sums by generalized Bernoulli numbers using only elementary methods from algebra and number theory.

Fibonacci And Lucas Differential Equations, Esra Erkus-Duman, Hakan Ciftci

Applications and Applied Mathematics: An International Journal (AAM)

The second-order linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the Fibonacci and the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions.

Simplifying Coefficients In A Family Of Ordinary Differential Equations Related To The Generating Function Of The Laguerre Polynomials, Feng Qi

Applications and Applied Mathematics: An International Journal (AAM)

In the paper, by virtue of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials.

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

Branch 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

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, 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, Dominic Klyve

Number Theory

No abstract provided.

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.

Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial …

Incomplete Generalized (P; Q; R)-Tribonacci Polynomials, Mark Shattuck, Elif Tan

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we consider an extension of the tribonacci polynomial, which we will refer to as the generalized (p; q; r)-tribonacci polynomial, denoted by T_n;m(x).We find an explicit formula for T_n;m(x)which we use to introduce the incomplete generalized (p; q; r)-tribonacci polynomials and derive several properties. An explicit formula for the generating function of the incomplete generalized polynomials is determined and a combinatorial interpretation is provided yielding further identities.

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 …

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 privacy-preserving, …

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 primary purpose …

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

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 unit …

On Rugina’S System Of Thought, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This article investigates Rugina's orientation table and gives particular examples for several of its seven models. Leon Walras's Economics of Stable Equilibrium and Keynes's Economics of Disequilibrium are combined in Rugina's orientation table in systems which are s percent stable and 100 ÿ s percent unstable, where s may be 100, 95, 65, 50, 35, 5, and 0. Classical logic and modern logic are united in Rugina's integrated logic, and then generalized in neutrosophic logic.

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 encryption, these …

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

Mathematics Publications

We give a list of PGL₂(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, 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.

A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin

Mathematics Sciences: Faculty Publications

In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.