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Articles 1 - 13 of 13
Full-Text Articles in Number Theory
A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn
A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn
Rose-Hulman Undergraduate Mathematics Journal
Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We …
Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas
Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas
Open Educational Resources
The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore logics, both propositional and first-order , and the construction of proofs, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through set theory. This theory is then put in practice in several applications. The particular (but quite widespread) case of equivalence and order relations is studied with detail. Then we introduces sequences and proofs by induction, followed by number theory. Finally, a small introduction to combinatorics is …
Algorithms Related To Triangle Groups, Bao The Pham
Algorithms Related To Triangle Groups, Bao The Pham
LSU Doctoral Dissertations
Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …
On The Mersenne Prime Numbers, Julia Vanlandingham
On The Mersenne Prime Numbers, Julia Vanlandingham
Undergraduate Honors Thesis Projects
The prime numbers have been an important field of research for thousands of years and are intertwined with most other fields of mathematics. One topic that has piqued the interest of mathematicians young and old is the Mersenne prime numbers, which have applications in many mathematics and computer science fields. The Mersenne primes get a lot of attention because there is not much known about them. However, we do have a very simple primality test for Mersenne numbers, which is why the largest currently known primes are Mersenne primes. These primes are also very closely related to another class of …
The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe
The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe
Celebration of Learning
A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.
Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke
Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke
Honors Theses
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 …
Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, Karthik Chandrasekhar
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 E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. 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 ( …
An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, Jacob Liddy
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.
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Senior Projects Spring 2016
Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group's irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs.
Solving Diophantine Equations, Florentin Smarandache, Octavian Cira
Solving Diophantine Equations, Florentin Smarandache, Octavian Cira
Branch Mathematics and Statistics Faculty and Staff Publications
In recent times, we witnessed an explosion of Number Theory problems that are solved using mathematical software and powerful computers. The observation that the number of transistors packed on integrated circuits doubles every two years made by Gordon E. Moore in 1965 is still accurate to this day. With ever increasing computing power more and more mathematical problems can be tacked using brute force. At the same time the advances in mathematical software made tools like Maple, Mathematica, Matlab or Mathcad widely available and easy to use for the vast majority of the mathematical research community. This tools don’t only …
Galois Representations From Non-Torsion Points On Elliptic Curves, Matthew Phillip Hughes
Galois Representations From Non-Torsion Points On Elliptic Curves, Matthew Phillip Hughes
Senior Projects Spring 2013
Working from well-known results regarding l-adic Galois representations attached to elliptic curves arising from successive preimages of the identity, we consider a natural deformation. Given a non-zero point P on a curve, we investigate the Galois action on the splitting fields of preimages of P under multiplication-by-l maps. We give a group-theoretic structure theorem for the corresponding Galois group, and state a conjecture regarding composita of two such splitting fields.
The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans
The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans
Mathematical Sciences Technical Reports (MSTR)
The discrete logarithm problem, and its adaptation to elliptic curves, called the elliptic curve discrete logarithm problem (ECDLP) is an open problem in the field of number theory, and its applications to modern cryptographic algorithms are numerous. This paper focuses on a statistical analysis of a modification to the ECDLP, called the x-ECDLP, where one is only given the xcoordinate of a point, instead of the entire point. Focusing only on elliptic curves whose field of definition is smaller than the number of points, this paper attempts to find a statistical indication of underlying structure (or lack thereof) in the …
Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly
Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly
Subhajit Kumar Ganguly
The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.