Primality Testing for Beginners

Author: Lasse Rempe-Gillen, Rebecca Waldecker
Publisher: American Mathematical Soc.
ISBN: 0821898833
Category: Mathematics
Page: 240
View: 7389
How can you tell whether a number is prime? What if the number has hundreds or thousands of digits? This question may seem abstract or irrelevant, but in fact, primality tests are performed every time we make a secure online transaction. In 2002, Agrawal, Kayal, and Saxena answered a long-standing open question in this context by presenting a deterministic test (the AKS algorithm) with polynomial running time that checks whether a number is prime or not. What is more, their methods are essentially elementary, providing us with a unique opportunity to give a complete explanation of a current mathematical breakthrough to a wide audience. Rempe-Gillen and Waldecker introduce the aspects of number theory, algorithm theory, and cryptography that are relevant for the AKS algorithm and explain in detail why and how this test works. This book is specifically designed to make the reader familiar with the background that is necessary to appreciate the AKS algorithm and begins at a level that is suitable for secondary school students, teachers, and interested amateurs. Throughout the book, the reader becomes involved in the topic by means of numerous exercises.

Applied Number Theory

Author: Harald Niederreiter,Arne Winterhof
Publisher: Springer
ISBN: 3319223216
Category: Mathematics
Page: 442
View: 9455
This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas. Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc. Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters 2-5 and offer a glimpse of advanced results that are presented without proofs and require more advanced mathematical skills. In the last chapter they review several further applications of number theory, ranging from check-digit systems to quantum computation and the organization of raster-graphics memory. Upper-level undergraduates, graduates and researchers in the field of number theory will find this book to be a valuable resource.

The Joy of Factoring

Author: Samuel S. Wagstaff (Jr.)
Publisher: American Mathematical Soc.
ISBN: 1470410486
Category: Mathematics
Page: 293
View: 2256
This book is about the theory and practice of integer factorisation presented in a historic perspective. It describes about twenty algorithms for factoring and a dozen other number theory algorithms that support the factoring algorithms. Most algorithms are described both in words and in pseudocode to satisfy both number theorists and computer scientists. Each of the ten chapters begins with a concise summary of its contents. The book starts with a general explanation of why factoring integers is important. The next two chapters present number theory results that are relevant to factoring. Further on there is a chapter discussing, in particular, mechanical and electronic devices for factoring, as well as factoring using quantum physics and DNA molecules. Another chapter applies factoring to breaking certain cryptographic algorithms. Yet another chapter is devoted to practical vs. theoretical aspects of factoring. The book contains more than 100 examples illustrating various algorithms and theorems. It also contains more than 100 interesting exercises to test the reader's understanding. Hints or answers are given for about a third of the exercises. The book concludes with a dozen suggestions of possible new methods for factoring integers. This book is written for readers who want to learn more about the best methods of factoring integers, many reasons for factoring, and some history of this fascinating subject. It can be read by anyone who has taken a first course in number theory.

Factorization and Primality Testing

Author: David M. Bressoud
Publisher: Springer Science & Business Media
ISBN: 1461245443
Category: Mathematics
Page: 240
View: 5145
"About binomial theorems I'm teeming with a lot of news, With many cheerful facts about the square on the hypotenuse. " - William S. Gilbert (The Pirates of Penzance, Act I) The question of divisibility is arguably the oldest problem in mathematics. Ancient peoples observed the cycles of nature: the day, the lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. Even when the inaccuracy of a 360-day year became apparent, they preferred to retain it and add five intercalary days. The number 360 retains its psychological appeal today because it is divisible by many small integers. The technical term for such a number reflects this appeal. It is called a "smooth" number. At the other extreme are those integers with no smaller divisors other than 1, integers which might be called the indivisibles. The mystic qualities of numbers such as 7 and 13 derive in no small part from the fact that they are indivisibles. The ancient Greeks realized that every integer could be written uniquely as a product of indivisibles larger than 1, what we appropriately call prime numbers. To know the decomposition of an integer into a product of primes is to have a complete description of all of its divisors.

Elementary Algebraic Geometry

Author: Klaus Hulek
Publisher: American Mathematical Soc.
ISBN: 0821829521
Category: Mathematics
Page: 213
View: 2173
This book is a true introduction to the basic concepts and techniques of algebraic geometry. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary.

Elliptic Curves, Modular Forms, and Their L-functions

Author: Alvaro Lozano-Robledo
Publisher: American Mathematical Soc.
ISBN: 0821852426
Category: Mathematics
Page: 195
View: 5382
Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

Ramsey Theory on the Integers

Second Edition
Author: Bruce M. Landman, Aaron Robertson
Publisher: American Mathematical Soc.
ISBN: 0821898671
Category: Mathematics
Page: 384
View: 5589
Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an "inequality" version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. Many new results and proofs have been added, most of which were not known when the first edition was published. Furthermore, the book's tables, exercises, lists of open research problems, and bibliography have all been significantly updated. This innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subject. This breakthrough book will engage students, teachers, and researchers alike.

Algebra für Einsteiger

Author: Jörg Bewersdorff
Publisher: American Mathematical Soc.
ISBN: 0821838172
Category: Mathematics
Page: 180
View: 5847
Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations. Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed.Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular $n$-gons are also presented. This book is suitable for undergraduates and beginning graduate students.

Problems in Abstract Algebra

Author: A. R. Wadsworth
Publisher: American Mathematical Soc.
ISBN: 1470435837
Category: Algebra, Abstract
Page: 277
View: 9719
This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.

McGraw-Hill's GED Mathematics

The Most Comprehensive and Reliable Study Program for the GED Math Test
Author: Jerry Howett
Publisher: McGraw Hill Professional
ISBN: 9780071407069
Category: Education
Page: 464
View: 1031
This focused format allows students to pretest for strengths and weaknesses in a given subject, thoroughly review core content areas, and finally check their exam-readiness with a full-length posttest in GED format. Covers basic operations through more complex activities for both the Casio FX-260 calculator and longhand problems.

GED Mathematical Reasoning Test For Dummies

Author: Murray Shukyn,Achim K. Krull
Publisher: John Wiley & Sons
ISBN: 1119030102
Category: Study Aids
Page: 240
View: 3106
Gear up to crush the GED Mathematical Test Does the thought of taking the GED Mathematical Reasoning Test make you weak? Fear not! With the help of GED Mathematical Reasoning Test For Dummies, you'll get up to speed on the new structure and computer-based format of the GED and gain the confidence and know-how to make the Mathematical Reasoning Test your minion. Packed with helpful guidance and instruction, this hands-on test-prep guide covers the concepts covered on the GED Mathematical Reasoning Test and gives you ample practice opportunities to assess your understanding of number operations/number sense, measurement and geometry, data, statistics, and probability, and algebra, functions, and patterns. Now a grueling 115 minutes long, the new Mathematical Reasoning section of the GED includes multiple choice, fill-in-the-blank, hot-spot, drop-down, and drag-and-drop questions—which can prove to be quite intimidating for the uninitiated. Luckily, this fun and accessible guide breaks down each section of the exam and the types of questions you'll encounter into easily digestible parts, making everything you'll come across on exam day feel like a breeze! Inside, you'll find methods to sharpen your math skills, tips on how to approach GED Mathematical Reasoning question types and formats, practice questions and study exercises, and a full-length practice test to help you pinpoint where you need more study help. Presents reviews of the GED Mathematical Reasoning test question types and basic computer skills Offers practice questions assessing work-place related and academic-based math skills Includes one full-length GED Mathematical Reasoning practice test Provides scoring guidelines and detailed answer explanations Even if math has always made you mad, GED Mathematical Reasoning Test For Dummies makes it easy to pass this crucial exam and obtain your hard-earned graduate equivalency diploma.

An Introduction to Number Theory with Cryptography

Author: James S. Kraft,Lawrence C. Washington
Publisher: CRC Press
ISBN: 1482214423
Category: Mathematics
Page: 572
View: 1328
Number theory has a rich history. For many years it was one of the purest areas of pure mathematics, studied because of the intellectual fascination with properties of integers. More recently, it has been an area that also has important applications to subjects such as cryptography. An Introduction to Number Theory with Cryptography presents number theory along with many interesting applications. Designed for an undergraduate-level course, it covers standard number theory topics and gives instructors the option of integrating several other topics into their coverage. The "Check Your Understanding" problems aid in learning the basics, and there are numerous exercises, projects, and computer explorations of varying levels of difficulty.

Officer Candidate Tests For Dummies

Author: Jane R. Burstein,Carolyn C. Wheater
Publisher: John Wiley & Sons
ISBN: 047059876X
Category: Education
Page: 456
View: 5998
Provides guidelines, tips, and advice for United States military officer examinations, including subject-specific exams and practice tests for officer careers in the Army, Air Force, Navy, Marine Corps, or Coast Guard.

ACT Math For Dummies

Author: Mark Zegarelli
Publisher: John Wiley & Sons
ISBN: 1118001540
Category: Education
Page: 360
View: 7780
Provides an easy-to-follow study guide for the ACT math test, complete with practice problems and strategies to help you prepare for exam day.

Mathematical Methods in Science

Author: George Pólya,Leon Bowden
Publisher: Cambridge University Press
ISBN: 9780883856260
Category: Mathematics
Page: 234
View: 8181
This book captures some of Pólya's excitement and vision. Its distinctive feature is the stress on the history of certain elementary chapters of science; these can be a source of enjoyment and deeper understanding of mathematics even for beginners who have little, or perhaps no, knowledge of physics.

Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Author: Thomas R. Shemanske
Publisher: American Mathematical Soc.
ISBN: 1470435829
Category: Cryptography
Page: 252
View: 9766
This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry combine to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bézout's theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard's method of factorization, Diffie–Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra's elliptic curve factorization method and ECC. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises further guide the exploration.

Praxis II Middle School Mathematics (5169) With Online Practice Tests

Author: Stephen Reiss
Publisher: Research & Education Association
ISBN: 9780738611846
Category: Education
Page: 320
View: 1026
REA's PRAXIS II: Middle School Mathematics Test Prep with Online Practice Gets You Certified and in the Classroom! Nationwide, more than 4 million teachers will be needed over the next decade, and all must take appropriate tests to be licensed. REA gets you ready for your teaching career with our outstanding library of Teacher Certification test preps. REA's PRAXIS II: Middle School Mathematics (5169) test prep is designed to help you master the information on this important exam, bringing you one step closer to being certified to teach middle school math. It's perfect for college students, out-of-state teachers, and career-changing professionals who are looking to teach middle school math in states where this exam is required. This study package contains in-depth reviews of all the content categories and topics tested on the PRAXIS II: Middle School Mathematics exam. A pretest and posttest are offered online to help you pinpoint strengths and weaknesses as you study the review chapters. The book includes two full-length practice tests based on actual PRAXIS questions and features every type of question, subject area, and skill tested on the exam. Two additional practice tests are offered online at the REA Study Center and come with automatic scoring, timed testing conditions, and diagnostic feedback to help you zero in on the topics and types of questions that give you trouble now, so you can succeed on test day. This test prep is a must-have for anyone prepping for the PRAXIS II: Middle School Math exam!

Discrete Mathematics and Its Applications

Author: Kenneth Rosen
Publisher: McGraw-Hill Higher Education
ISBN: 007741893X
Category: Education
Page: N.A
View: 2868

Library recommendations for undergraduate mathematics

Author: Lynn Arthur Steen,Mathematical Association of America
Publisher: Mathematical Assn of Amer
Category: Mathematics
Page: 194
View: 4773

The Official ACT Prep Guide

Author: ACT
Publisher: John Wiley & Sons
ISBN: 1119398908
Category: Study Aids
Page: 720
View: 4757
The only guide from the ACT organization, the makers of the exam, revised and updated for 2017 and beyond The Official ACT Prep Guide, 2018 Edition, Revised and Updated is the must-have resource for college bound students. The guide is the go-to handbook for ACT preparation and the only guide from the makers of the exam. The book and online content includes the actual ACT test forms (taken from real ACT exams). In addition, this comprehensive resource has everything students need to know about when they are preparing for and taking the ACT. The book contains information on how to register for the exam, proven test-taking strategies, ideas for preparing mentally and physically, gearing up for test day, and much more. This invaluable guide includes additional questions and material that contains articles on everything from preparing a standout college application and getting into your top-choice school to succeeding in college The bestselling prep guide from the makers of the ACT test Offers bonus online contest to help boost college readiness Contains the real ACT test forms used in previous years This new edition offers students updated data on scoring your writing test, new reporting categories, as well as updated tips on how to do your best preparing for the test and on the actual test day from the team at ACT. It also offers additional 400 practice questions that are available online.