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| basic number theory problems and solutions: Methods of Solving Number Theory Problems Ellina Grigorieva, 2018-07-06 Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat’s (Pell’s) equations. It also covers Fermat’s factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring’s problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence. |
| basic number theory problems and solutions: Problems in Algebraic Number Theory M. Ram Murty, Jody Esmonde, 2005 The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved |
| basic number theory problems and solutions: 250 Problems in Elementary Number Theory Wacław Sierpiński, 1970 |
| basic number theory problems and solutions: Unsolved Problems in Number Theory Richard Guy, 2013-11-11 To many laymen, mathematicians appear to be problem solvers, people who do hard sums. Even inside the profession we dassify ouselves as either theorists or problem solvers. Mathematics is kept alive, much more than by the activities of either dass, by the appearance of a succession of unsolved problems, both from within mathematics itself and from the increasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solution of a problem may stifte interest in the area around it. But Fermat 's Last Theorem, because it is not yet a theorem, has generated a great deal of good mathematics, whether goodness is judged by beauty, by depth or by applicability. To pose good unsolved problems is a difficult art. The balance between triviality and hopeless unsolvability is delicate. There are many simply stated problems which experts tell us are unlikely to be solved in the next generation. But we have seen the Four Color Conjecture settled, even if we don't live long enough to learn the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfect numbers. On the other hand, unsolved problems may not be unsolved at all, or than was at first thought. |
| basic number theory problems and solutions: Elementary Number Theory Underwood Dudley, 2012-06-04 Written in a lively, engaging style by the author of popular mathematics books, this volume features nearly 1,000 imaginative exercises and problems. Some solutions included. 1978 edition. |
| basic number theory problems and solutions: Problems Of Number Theory In Mathematical Competitions Hong-bing Yu, 2009-09-16 Number theory is an important research field of mathematics. In mathematical competitions, problems of elementary number theory occur frequently. These problems use little knowledge and have many variations. They are flexible and diverse. In this book, the author introduces some basic concepts and methods in elementary number theory via problems in mathematical competitions. Readers are encouraged to try to solve the problems by themselves before they read the given solutions of examples. Only in this way can they truly appreciate the tricks of problem-solving. |
| basic number theory problems and solutions: Elementary Number Theory: Primes, Congruences, and Secrets William Stein, 2008-10-28 This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predeterminedsecret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem. |
| basic number theory problems and solutions: Mathematical Problems and Proofs Branislav Kisacanin, 2007-05-08 A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm. This excellent primer illustrates more than 150 solutions and proofs, thoroughly explained in clear language. The generous historical references and anecdotes interspersed throughout the text create interesting intermissions that will fuel readers' eagerness to inquire further about the topics and some of our greatest mathematicians. The author guides readers through the process of solving enigmatic proofs and problems, and assists them in making the transition from problem solving to theorem proving. At once a requisite text and an enjoyable read, Mathematical Problems and Proofs is an excellent entrée to discrete mathematics for advanced students interested in mathematics, engineering, and science. |
| basic number theory problems and solutions: Problem-Solving and Selected Topics in Number Theory Michael Th. Rassias, 2010-11-16 The book provides a self-contained introduction to classical Number Theory. All the proofs of the individual theorems and the solutions of the exercises are being presented step by step. Some historical remarks are also presented. The book will be directed to advanced undergraduate, beginning graduate students as well as to students who prepare for mathematical competitions (ex. Mathematical Olympiads and Putnam Mathematical competition). |
| basic number theory problems and solutions: A Friendly Introduction to Number Theory Joseph H. Silverman, 2013-10-03 For one-semester undergraduate courses in Elementary Number Theory. A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet—number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results. |
| basic number theory problems and solutions: Solved and Unsolved Problems in Number Theory Daniel Shanks, 2024-01-24 The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers. |
| basic number theory problems and solutions: 111 Problems in Algebra and Number Theory Adrian Andreescu, Vinjai Vale, 2016 Algebra plays a fundamental role not only in mathematics, but also in various other scientific fields. Without algebra there would be no uniform language to express concepts such as numbers' properties. Thus one must be well-versed in this domain in order to improve in other mathematical disciplines. We cover algebra as its own branch of mathematics and discuss important techniques that are also applicable in many Olympiad problems. Number theory too relies heavily on algebraic machinery. Often times, the solutions to number theory problems involve several steps. Such a solution typically consists of solving smaller problems originating from a hypothesis and ending with a concrete statement that is directly equivalent to or implies the desired condition. In this book, we introduce a solid foundation in elementary number theory, focusing mainly on the strategies which come up frequently in junior-level Olympiad problems. |
| basic number theory problems and solutions: The USSR Olympiad Problem Book D. O. Shklarsky, N. N. Chentzov, I. M. Yaglom, 2013-04-15 Over 300 challenging problems in algebra, arithmetic, elementary number theory and trigonometry, selected from Mathematical Olympiads held at Moscow University. Only high school math needed. Includes complete solutions. Features 27 black-and-white illustrations. 1962 edition. |
| basic number theory problems and solutions: Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall, 2001-12-12 First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem. Intended as a upper level textbook, it |
| basic number theory problems and solutions: Number Theory George E. Andrews, 2012-04-30 Undergraduate text uses combinatorial approach to accommodate both math majors and liberal arts students. Covers the basics of number theory, offers an outstanding introduction to partitions, plus chapters on multiplicativity-divisibility, quadratic congruences, additivity, and more. |
| basic number theory problems and solutions: Elementary Number Theory Gareth A. Jones, Josephine M. Jones, 2012-12-06 An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles. |
| basic number theory problems and solutions: A Brief Guide to Algebraic Number Theory H. P. F. Swinnerton-Dyer, 2001-02-22 Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author. |
| basic number theory problems and solutions: Number Theory Kuldeep Singh, 2020 Number Theory: Step by Step is an undergraduate-level introduction to number theory that assumes no prior knowledge, but works to gradually increase the reader's confidence and ability to tackle more difficult number theory material. |
| basic number theory problems and solutions: Discrete Mathematics and Its Applications Kenneth Rosen, 2006-07-26 Discrete Mathematics and its Applications, Sixth Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over 500 institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications...from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields. |
| basic number theory problems and solutions: The Theory of Numbers Andrew Adler, John E. Coury, 1995 |
| basic number theory problems and solutions: Introduction to Number Theory Anthony Vazzana, Martin Erickson, David Garth, 2007-10-30 One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topi |
| basic number theory problems and solutions: Abstract Algebra Thomas Judson, 2023-08-11 Abstract Algebra: Theory and Applications is an open-source textbook that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many non-trivial applications. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory. |
| basic number theory problems and solutions: 1001 Problems in Classical Number Theory J. M. de Koninck, Armel Mercier, 2007 In the spirit of The Book of the One Thousand and One Nights, the authors offer 1001 problems in number theory in a way that entices the reader to immediately attack the next problem. Whether a novice or an experienced mathematician, anyone fascinated by numbers will find a great variety of problems--some simple, others more complex--that will provide them with a wonderful mathematical experience. |
| basic number theory problems and solutions: 100 Great Problems of Elementary Mathematics Heinrich Dörrie, 2013-04-09 Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge, Steiner, and other great mathematical minds. Features squaring the circle, pi, and similar problems. No advanced math is required. Includes 100 problems with proofs. |
| basic number theory problems and solutions: Number Theory Róbert Freud, Edit Gyarmati, 2020-10-08 Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise. The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are well-known mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors' personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers. |
| basic number theory problems and solutions: Old and New Unsolved Problems in Plane Geometry and Number Theory Victor Klee, Stan Wagon, 1991-12-31 |
| basic number theory problems and solutions: An Illustrated Theory of Numbers Martin H. Weissman, 2017 Seeing arithmetic -- Foundations -- The Euclidean algorithm -- Prime factorization -- Rational and constructible numbers -- Gaussian and Eisenstein integers -- Modular arithmetic -- The modular worlds -- Modular dynamics -- Assembling the modular worlds -- Quadratic residues -- Quadratic forms -- The topograph -- Definite forms -- Indefinite forms |
| basic number theory problems and solutions: The Stanford Mathematics Problem Book George Polya, Jeremy Kilpatrick, 2013-04-09 Based on Stanford University's well-known competitive exam, this excellent mathematics workbook offers students at both high school and college levels a complete set of problems, hints, and solutions. 1974 edition. |
| basic number theory problems and solutions: Problem-Solving Strategies Arthur Engel, 2008-01-19 A unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a problem of the week, thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market. |
| basic number theory problems and solutions: Introduction to Analytic Number Theory Tom M. Apostol, 1998-05-28 This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages.-—MATHEMATICAL REVIEWS |
| basic number theory problems and solutions: Number Fields Daniel A. Marcus, 2018-07-05 Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises. |
| basic number theory problems and solutions: Elementary Number Theory Joe Roberts, 1925 |
| basic number theory problems and solutions: Elements of Number Theory John Stillwell, 2002-12-13 Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement. |
| basic number theory problems and solutions: Elementary Number Theory with Programming Marty Lewinter, Jeanine Meyer, 2015-05-06 A highly successful presentation of the fundamental concepts of number theory and computer programming Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and concepts in either area. Elementary Number Theory with Programming features comprehensive coverage of the methodology and applications of the most well-known theorems, problems, and concepts in number theory. Using standard mathematical applications within the programming field, the book presents modular arithmetic and prime decomposition, which are the basis of the public-private key system of cryptography. In addition, the book includes: Numerous examples, exercises, and research challenges in each chapter to encourage readers to work through the discussed concepts and ideas Select solutions to the chapter exercises in an appendix Plentiful sample computer programs to aid comprehension of the presented material for readers who have either never done any programming or need to improve their existing skill set A related website with links to select exercises An Instructor’s Solutions Manual available on a companion website Elementary Number Theory with Programming is a useful textbook for undergraduate and graduate-level students majoring in mathematics or computer science, as well as an excellent supplement for teachers and students who would like to better understand and appreciate number theory and computer programming. The book is also an ideal reference for computer scientists, programmers, and researchers interested in the mathematical applications of programming. |
| basic number theory problems and solutions: Primes of the Form X2 + Ny2 David A. Cox, 1989-09-28 Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively. |
| basic number theory problems and solutions: A Problem Based Journey from Elementary Number Theory to an Introduction to Matrix Theory Abraham Berman, 2021-10-18 The book is based on lecture notes of a course 'from elementary number theory to an introduction to matrix theory' given at the Technion to gifted high school students. It is problem based, and covers topics in undergraduate mathematics that can be introduced in high school through solving challenging problems. These topics include Number theory, Set Theory, Group Theory, Matrix Theory, and applications to cryptography and search engines. |
| basic number theory problems and solutions: Elementary Introduction to Number Theory Calvin T. Long, 1965 |
| basic number theory problems and solutions: Elementary School Math Contests Steven Doan, Jesse Doan, 2017-08-15 Elementary School Math Contests contains over 500 challenging math contest problems and detailed step-by-step solutions in Number Theory, Algebra, Counting & Probability, and Geometry. The problems and solutions are accompanied with formulas, strategies, and tips.This book is written for beginning mathletes who are interested in learning advanced problem solving and critical thinking skills in preparation for elementary and middle school math competitions. |
| basic number theory problems and solutions: Notes and Problems in Number Theory (Volume II) Taha Sochi, 2024-05-23 This is the second volume of my book Notes and Problems in Number Theory. We focus in this volume on the type of problems that develop the basic and most essential skills which are required for dealing with number theory problems. We introduced some new topics in the first chapter (i.e. Introduction), while the remaining chapters are largely dedicated to solved problems from the main topics of elementary number theory (which are introduced in V1 or in the Introduction chapter of the present volume). We also introduced the subject of cryptography and computing in number theory in the last two chapters. So in brief, the materials in this volume are largely a mix of applications to the materials of V1 and some theoretical background of new topics as well as applications to the new topics. As in my previous books, my topmost priority in the structure and presentation is clarity and graduality so that the readers have the best chance of understanding the content with minimum effort and with maximum enjoyment. The book can be used as a text or as a reference for an introductory course on number theory and may also be used for general reading in mathematics (especially by those who have the hobby of problem solving). The book may also be adopted as a source of pedagogical materials which can supplement, for instance, tutorial sessions (e.g. in undergraduate courses on mathematics or computing or cryptography or related subjects). |
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为什么说以Basic作为入门语言会变成脑残？ - 知乎
Dijkstra说的这个basic是上古时期的basic，参考小霸王上的basic。其中充斥着GOTO，每行必须有行号，行号满了就不能插入，变量命名受限，没有指针和动态内存分配，有很多使其无法胜任 …
base，basic，basis这个三个词怎么区分? - 知乎
Aug 7, 2020 · basic（尤指作为发展的起点）基本的，初步的，如： 6. He doesn't have mastery of the basic skills of reading, writing and communicating. 他还没掌握基本的读写和交流技巧。【 …
excel2021visual basic打开是灰色的怎么办？ - 知乎
如果Excel 2021 中的 Visual Basic 编辑器打开时显示为灰色，可能是由于以下原因之一： 安装问题：确保已正确安装了 Visual Basic for Applications（VBA）组件。 检查 Microsoft Office 安 …
一文了解Transformer全貌（图解Transformer） - 知乎
Jan 21, 2025 · Transformer整体结构（输入两个单词的例子） 为了能够对Transformer的流程有个大致的了解，我们举一个简单的例子，还是以之前的为例，将法语"Je suis etudiant"翻译成英 …
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我的毕业论文参考文献中有大量的[sl]和[sn]标志,前者是表示出版地未知，后者是表示出版商未知，我需要高…
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个人4盘位NAS，用什么RAID比较合适，为什么？ - 知乎
两盘位basic：存放电影，下载，电脑备份等非重要数据。 可扩展一盘位usb外接（可以用电脑替代，更理想情况是有第二台nas）：使用套件做最重要的数据定期同步或备份，电影种子，basic …
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如何origin在一个图中画两条线，比如这种? - 知乎
导入数据到各个列中，全选数据后，点击 Origin 工具栏上的 Plot ——> Basic 2D ——> Line + Symbol 或者 点击 Origin 下边快捷图标 ，如下图所示 Origin 就会自动绘制两条数据线，如下图 …
