Number Theory Commons^™

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

Scholarship

style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;"> prime factors for every style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;"> between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes style="color: rgb(51, 51, 51); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;">with the property that …

Eccentricity, Space Bending, Dimension, Florentin Smarandache, Marian Nitu, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

The main goal of this paper is to present new transformations, previously non-existent in traditional mathematics, that we call centric mathematics (CM) but that became possible due to the new born eccentric mathematics, and, implicitly, to the supermathematics (SM).

As shown in this work, the new geometric transformations, namely conversion or transfiguration, wipe the boundaries between discrete and continuous geometric forms, showing that the first ones are also continuous, being just apparently discontinuous.

On Crittenden And Vanden Eynden's Conjecture, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

It is possible to cover all (positive) integers with n geometrical progressions of integers? Find a necessary and sufficient condition for a general class of positive integer sequences such that, for a fixed n , there are n (distinct) sequences of this class which cover all integers.

Neutrosophic Crisp Sets & Neutrosophic Crisp Topological Spaces, A. A. Salama, Florentin Smarandache, Valeri Kroumov

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we generalize the crisp topological space to the notion of neutrosophic crisp topological space, and we construct the basic concepts of the neutrosophic crisp topology. In addition to these, we introduce the definitions of neutrosophic crisp continuous function and neutrosophic crisp compact spaces. Finally, some characterizations concerning neutrosophic crisp compact spaces are presented and one obtains several properties. Possible application to GIS topology rules are touched upon.

Lower And Upper Soft Interval Valued Neutrosophic Rough Approximations Of An Ivnss-Relation, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we extend the lower and upper soft interval valued intuitionistic fuzzy rough approximations of IVIFS –relations proposed by Anjan et al. to the case of interval valued neutrosophic soft set relation(IVNSS-relation for short)

Intuitionistic Neutrosophic Soft Set Over Rings, Florentin Smarandache, Said Broumi, Pabitra Kumar Maji

Branch Mathematics and Statistics Faculty and Staff Publications

S.Broumi and F.Smarandache introduced the concept of intuitionistic neutrosophic soft set as an extension of the soft set theory. In this paper we have applied the concept of intuitionistic neutrosophic soft set to rings theory .The notion of intuitionistic neutrosophic soft set over ring (INSSOR for short ) is introduced and their basic properties have been investigated.The definitions of intersection, union, AND, and OR operations over ring (INSSOR) have also been defined. Finally, we have defined the product of two intuitionistic neutrosophic soft set over ring.

Single Valued Neutrosophic Information Systems Based On Rough Set Theory, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The theory of rough sets was firstly proposed by Pawlak. Later on, Smarandache introduced the concept of neutrosophic (NS) sets in 1998. In this paper based on the concept of rough neutrosohic set, we define the concept of single valued neutrosophic information systems. In addition, we will discuss the knowledge reduction and extension of the single valued neutrosophic information systems.

Interval Neutrosophic Rough Set, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper combines interval-valued neutrosophic sets and rough sets. It studies rougheness in interval-valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower an upper approximations of interval neutrosophic sets.

Interval Neutrosophic Logic, Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman

Branch Mathematics and Statistics Faculty and Staff Publications

Interval Neutrosophic Logic

Importance Of Sources Using The Repeated Fusion Method And The Proportional Conflict Redistribution Rules #5 And #6, Florentin Smarandache, Jean Dezert

Branch Mathematics and Statistics Faculty and Staff Publications

We present in this paper some examples of how to compute by hand the PCR5 fusion rule for three sources, so the reader will better understand its mechanism. We also take into consideration the importance of sources, which is different from the classical discounting of sources.

Algebraic Structures On The Fuzzy Interval [0, 1), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup. The semigroup under × has no finite subsemigroups but under min or max we have subsemigroups of order one, two and so on. [0, 1) under + …

Special Pseudo Linear Algebras Using [0,N), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce some special type of linear algebras called pseudo special linear algebras using the interval [0, n). These new types of special pseudo interval linear algebras has several interesting properties. Special pseudo interval linear algebras are built over the subfields in Zn where Zn is a S-ring. We study the substructures of them. The notion of Smarandache special interval pseudo linear algebras and Smarandache strong special pseudo interval linear algebras are introduced. The former Sspecial interval pseudo linear algebras are built over the Sring itself. Study in this direction has yielded several interesting results. S-strong special …

Algebraic Structures On Finite Complex Modulo Integer Interval C([0, N)), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers was introduced by the authors in 2011. Now using this finite complex modulo integer intervals several algebraic structures are built. Further the concept of finite complex modulo integers itself happens to be new and innovative for in case of finite complex modulo integers the square value of the finite complex number varies with varying n of Zn. In case of finite complex modulo integer intervals also we can have only pseudo ring as the distributive law is not true, in general in C([0, …

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

New Research On Neutrosophic Algebraic Structures, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “ I “ gives rise to a more bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories: such as neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical …

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 …

The Kronecker-Weber Theorem: An Exposition, Amber Verser

Lawrence University Honors Projects

This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in The American Mathematical Monthly in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number …

Quotients Of Gaussian Primes, Stephan Ramon Garcia

Pomona Faculty Publications and Research

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"

Rock Art Tallies: Mathematics On Stone In Western North America, James V. Rauff

Journal of Humanistic Mathematics

Western North America abounds with rock art sites. From Alberta to New Mexico and from Minnesota to California one can find the enigmatic rock paintings and rock carvings left by the pre-Columbian inhabitants. The images left behind on the rocks of the American plains and deserts are those of humanoids and animals, arrows and spears, and a variety of geometric shapes and abstract designs. Also included, in great numbers, are sequences of repeated shapes and marks that scholars have termed "tallies." The tallies are presumed to be an ancient accounting of something or some things. This article examines rock art …

Number Theory Applications In Cryptography, Francesca Pizzigoni

Theses, Dissertations and Culminating Projects

This thesis provides a unique cryptosystem comprised of different number theory applications. We first consider the well-known Knapsack Problem and the resulting Knapsack Cryptosystem. It is known that when the Knapsack Problem involves a superincreasing sequence, the solution is easy to find. Two cryptosystems are designed and displayed in this thesis that allow two parties often called Alice and Bob use a common superincreasing sequence in the encryption and decryption process. They use this sequence and a variation of the Knapsack Cryptosystem to send and receive binary messages. The first cryptosystem assumes that Alice and Bob agree on a shared …