Portland State University
University of New Mexico
California State University, San Bernardino
New Mexico State University
Montclair State University
Benedictine University
Eastern Illinois University
The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security,
2023
William & Mary
The Vulnerabilities To The Rsa Algorithm And Future Alternative Algorithms To Improve Security, James Johnson
Cybersecurity Undergraduate Research Showcase
The RSA encryption algorithm has secured many large systems, including bank systems, data encryption in emails, several online transactions, etc. Benefiting from the use of asymmetric cryptography and properties of number theory, RSA was widely regarded as one of most difficult algorithms to decrypt without a key, especially since by brute force, breaking the algorithm would take thousands of years. However, in recent times, research has shown that RSA is getting closer to being efficiently decrypted classically, using algebraic methods, (fully cracked through limited bits) in which elliptic-curve cryptography has been thought of as the alternative that is stronger than …
Further Generalizations Of Happy Numbers,
2023
Bryn Mawr College
Further Generalizations Of Happy Numbers, E. Simonton Williams
Rose-Hulman Undergraduate Mathematics Journal
A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …
Divisibility Probabilities For Products Of Randomly Chosen Integers,
2023
University of Maryland
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Rose-Hulman Undergraduate Mathematics Journal
We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
Rough Numbers And Variations On The Erdős--Kac Theorem,
2023
Dartmouth College
Rough Numbers And Variations On The Erdős--Kac Theorem, Kai Fan
Dartmouth College Ph.D Dissertations
The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory.
In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y>=1. Denoting by Phi(x,y) the count of these numbers up to any given x>=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)<0.6x/\log y holds uniformly for all 3<=y<=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges.
In the second part, we turn …
On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields,
2023
The Graduate Center, City University of New York
On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields, Owen Sweeney
Dissertations, Theses, and Capstone Projects
We obtain a new simpler sufficient condition for Kolyvagin's criteria, regarding the second case of Fermat's last theorem with prime exponent p over the p-th cyclotomic field, to hold. It covers cases when the existing simpler sufficient conditions do not hold and is important for the theoretical study of the criteria.
On The Order-Type Complexity Of Words, And Greedy Sidon Sets For Linear Forms,
2023
The Graduate Center, City University of New York
On The Order-Type Complexity Of Words, And Greedy Sidon Sets For Linear Forms, Yin Choi Cheng
Dissertations, Theses, and Capstone Projects
This work consists of two parts. In the first part, we study the order-type complexity of right-infinite words over a finite alphabet, which is defined to be the order types of the set of shifts of said words in lexicographical order. The set of shifts of any aperiodic morphic words whose first letter in the purely-morphic pre-image occurs at least twice in the pre-image has the same order type as Q ∩ (0, 1), Q ∩ (0, 1], or Q ∩ [0, 1). This includes all aperiodic purely-morphic binary words. The order types of uniform-morphic ternary words were also studied, …
On The Spectrum Of Quaquaversal Operators,
2023
The Graduate Center, City University of New York
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
Dissertations, Theses, and Capstone Projects
In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …
Approaches To The Erdős–Straus Conjecture,
2023
CUNY New York City College of Technology
Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov
Publications and Research
The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring,
2023
Clemson University
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
All Dissertations
Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …
Zeros Of Modular Forms,
2023
Clemson University
Zeros Of Modular Forms, Daozhou Zhu
All Dissertations
Let $E_k(z)$ be the normalized Eisenstein series of weight $k$ for the full modular group $\text{SL}(2, \mathbb{Z})$. It is conjectured that all the zeros of the weight $k+\ell$ cusp form $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. Reitzes, Vulakh and Young \cite{Reitzes17} proved that this statement is true for sufficiently large $k$ and $\ell$. Xue and Zhu \cite{Xue} proved the cases when $\ell=4,6,8$ with $k\geq\ell$, all the zeros of $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ lie on the arc $|z|=1$. For all $k\geq\ell\geq10$, we will use the same method as \cite{Reitzes17} to locate these zeros on the arc $|z|=1$, and improve the …
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups,
2023
College of Saint Benedict/Saint John's University
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.
One Formula For Non-Prime Numbers: Motivations And Characteristics,
2023
Department of Basic Science, Faculty of Engineering, The British University in Egypt
One Formula For Non-Prime Numbers: Motivations And Characteristics, Mahmoud Mansour, Kamal Hassan Prof.
Basic Science Engineering
Primes are essential for computer encryption and cryptography, as they are fundamental units of whole numbers and are of the highest importance due to their mathematical qualities. However, identifying a pattern of primes is not easy. Thinking in a different way may get benefits, by considering the opposite side of the problem which means focusing on non-prime numbers. Recently, researchers introduced, the pattern of non-primes in two maximal sets while in this paper, non-primes are presented in one formula. Getting one-way formula for non-primes may pave the way for further applications based on the idea of primes.
Lagrange’S Study Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Study Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of The Converse Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of Wilson’S Theorem—And More!,
2023
Ursinus College
Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Alternate Proof Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem,
2023
Case Western Reserve University
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
Decomposition Of Beatty And Complementary Sequences,
2023
Smith College
Decomposition Of Beatty And Complementary Sequences, Geremías Polanco
Mathematics and Statistics: Faculty Publications
In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.
Coefficients Of Modular Forms And Applications To Partition Theory,
2023
University of Tennessee
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Doctoral Dissertations
We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves,
2023
University of Tennessee, Knoxville
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein
Doctoral Dissertations
Given an ordinary elliptic curve E over a field 𝕜 of characteristic p, there is an elliptic curve E over the Witt vectors W(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over 𝔽_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …
