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| number theory problems with 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. |
| number theory problems with solutions: Problems in Algebraic Number Theory M. Ram Murty, Jody (Indigo) Esmonde, 2005-09-28 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 |
| number theory problems with solutions: Number Theory Titu Andreescu, Dorin Andrica, 2009-06-12 This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for solving. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems. By emphasizing examples and applications the authors motivate and engage readers. |
| number theory problems with solutions: Problems of Number Theory in Mathematical Competitions Hong-Bing Yu, 2010 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. |
| number theory problems with solutions: The Theory of Numbers Andrew Adler, John E. Coury, 1995 |
| number theory problems with solutions: Introduction to Number Theory Mathew Crawford, 2008 Learn the fundamentals of number theory from former MATHCOUNTS, AHSME, and AIME perfect scorer Mathew Crawford. Topics covered in the book include primes & composites, multiples & divisors, prime factorization and its uses, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and much more. The text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes motivated solutions to these problems, through which concepts and curriculum of number theory are taught. Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains hundreds of problems ... This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of number theory will find this book an instrumental part of their mathematics libraries.--Publisher's website |
| number theory problems with solutions: Number Theory Titu Andreescu, Gabriel Dospinescu, Oleg Mushkarov, 2017-07-15 Challenge your problem-solving aptitude in number theory with powerful problems that have concrete examples which reflect the potential and impact of theoretical results. Each chapter focuses on a fundamental concept or result, reinforced by each of the subsections, with scores of challenging problems that allow you to comprehend number theory like never before. All students and coaches wishing to excel in math competitions will benefit from this book as will mathematicians and adults who enjoy interesting mathematics. |
| number theory problems with solutions: Elementary Number Theory Joe Roberts, 1925 |
| number theory problems with 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. |
| number theory problems with solutions: Unsolved Problems in Number Theory Richard Guy, R.K. Guy, 2013-06-29 Second edition sold 2241 copies in N.A. and 1600 ROW. New edition contains 50 percent new material. |
| number theory problems with 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 |
| number theory problems with 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 predetermined secret; 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. |
| number theory problems with solutions: 1001 Problems in Classical Number Theory Armel Mercier, 2007 |
| number theory problems with solutions: Number Theory Kuldeep Singh, 2020-10-08 Number theory is one of the oldest branches of mathematics that is primarily concerned with positive integers. While it has long been studied for its beauty and elegance as a branch of pure mathematics, it has seen a resurgence in recent years with the advent of the digital world for its modern applications in both computer science and cryptography. 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 material. The strength of the text is in its large number of examples and the step-by-step explanation of each topic as it is introduced to help aid understanding the abstract mathematics of number theory. It is compiled in such a way that allows self-study, with explicit solutions to all the set of problems freely available online via the companion website. Punctuating the text are short and engaging historical profiles that add context for the topics covered and provide a dynamic background for the subject matter. |
| number theory problems with 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. |
| number theory problems with 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. |
| number theory problems with solutions: Equations and Inequalities Jiri Herman, Radan Kucera, Jaromir Simsa, 2012-12-06 A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises. |
| number theory problems with solutions: 250 Problems in Elementary Number Theory Wacław Sierpiński, Waclaw Sierpinski, 1970 |
| number theory problems with 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. |
| number theory problems with solutions: Number Theory , 1986-05-05 This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences. |
| number theory problems with solutions: Problems And Solutions In Real Analysis (Second Edition) Masayoshi Hata, 2016-12-12 This second edition introduces an additional set of new mathematical problems with their detailed solutions in real analysis. It also provides numerous improved solutions to the existing problems from the previous edition, and includes very useful tips and skills for the readers to master successfully. There are three more chapters that expand further on the topics of Bernoulli numbers, differential equations and metric spaces.Each chapter has a summary of basic points, in which some fundamental definitions and results are prepared. This also contains many brief historical comments for some significant mathematical results in real analysis together with many references.Problems and Solutions in Real Analysis can be treated as a collection of advanced exercises by undergraduate students during or after their courses of calculus and linear algebra. It is also instructive for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the Prime Number Theorem through several exercises. This volume is also suitable for non-experts who wish to understand mathematical analysis. |
| number theory problems with solutions: Elementary Number Theory with Applications Thomas Koshy, 2007-05-08 This second edition updates the well-regarded 2001 publication with new short sections on topics like Catalan numbers and their relationship to Pascal's triangle and Mersenne numbers, Pollard rho factorization method, Hoggatt-Hensell identity. Koshy has added a new chapter on continued fractions. The unique features of the first edition like news of recent discoveries, biographical sketches of mathematicians, and applications--like the use of congruence in scheduling of a round-robin tournament--are being refreshed with current information. More challenging exercises are included both in the textbook and in the instructor's manual. Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and in-service math teachers at the high school and middle school levels. * Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises * Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes * Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle East |
| number theory problems with solutions: Elementary Number Theory James S. Kraft, Lawrence C. Washington, 2014-11-24 Elementary Number Theory takes an accessible approach to teaching students about the role of number theory in pure mathematics and its important applications to cryptography and other areas. The first chapter of the book explains how to do proofs and includes a brief discussion of lemmas, propositions, theorems, and corollaries. The core of the text covers linear Diophantine equations; unique factorization; congruences; Fermat’s, Euler’s, and Wilson’s theorems; order and primitive roots; and quadratic reciprocity. The authors also discuss numerous cryptographic topics, such as RSA and discrete logarithms, along with recent developments. The book offers many pedagogical features. The check your understanding problems scattered throughout the chapters assess whether students have learned essential information. At the end of every chapter, exercises reinforce an understanding of the material. Other exercises introduce new and interesting ideas while computer exercises reflect the kinds of explorations that number theorists often carry out in their research. |
| number theory problems with solutions: Steps into Analytic Number Theory Paul Pollack, Akash Singha Roy, 2021-02-08 This problem book gathers together 15 problem sets on analytic number theory that can be profitably approached by anyone from advanced high school students to those pursuing graduate studies. It emerged from a 5-week course taught by the first author as part of the 2019 Ross/Asia Mathematics Program held from July 7 to August 9 in Zhenjiang, China. While it is recommended that the reader has a solid background in mathematical problem solving (as from training for mathematical contests), no possession of advanced subject-matter knowledge is assumed. Most of the solutions require nothing more than elementary number theory and a good grasp of calculus. Problems touch at key topics like the value-distribution of arithmetic functions, the distribution of prime numbers, the distribution of squares and nonsquares modulo a prime number, Dirichlet's theorem on primes in arithmetic progressions, and more. This book is suitable for any student with a special interest in developing problem-solving skills in analytic number theory. It will be an invaluable aid to lecturers and students as a supplementary text for introductory Analytic Number Theory courses at both the undergraduate and graduate level. |
| number theory problems with 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. |
| number theory problems with solutions: Elements of Number Theory I. M. Vinogradov, 2016-01-14 Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition. |
| number theory problems with solutions: Introduction to Analytic Number Theory Tom M. Apostol, 2013-06-29 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 |
| number theory problems with solutions: A Course in Number Theory H. E. Rose, 1995 This textbook covers the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. This book excludes topics relying heavily on complex analysis and advanced algebraic number theory. The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on Fermat's last theorem is also briefly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables. |
| number theory problems with solutions: Number Theory and Its History Oystein Ore, 2012-07-06 Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography. |
| number theory problems with solutions: Discrete Mathematics Oscar Levin, 2016-08-16 This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the introduction to proof course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions. |
| number theory problems with solutions: An Illustrated Theory of Numbers Martin H. Weissman, 2020-09-15 News about this title: — Author Marty Weissman has been awarded a Guggenheim Fellowship for 2020. (Learn more here.) — Selected as a 2018 CHOICE Outstanding Academic Title — 2018 PROSE Awards Honorable Mention An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject. |
| number theory problems with solutions: 104 Number Theory Problems Titu Andreescu, Dorin Andrica, Zuming Feng, 2007-04-05 This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and in mathematical research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory. |
| number theory problems with 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. |
| number theory problems with solutions: Elements of Number Theory John Stillwell, 2012-11-12 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. |
| number theory problems with 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. |
| number theory problems with solutions: Winning Solutions Edward Lozansky, Cecil Rousseau, 2012-12-06 This book provides the mathematical tools and problem-solving experience needed to successfully compete in high-level problem solving competitions. Each section presents important background information and then provides a variety of worked examples and exercises to help bridge the gap between what the reader may already know and what is required for high-level competitions. Answers or sketches of the solutions are given for all exercises. |
| number theory problems with 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 |
| number theory problems with solutions: A Selection of Problems in the Theory of Numbers Waclaw Sierpinski, 2014-05-16 A Selection of Problems in the Theory of Numbers focuses on mathematical problems within the boundaries of geometry and arithmetic, including an introduction to prime numbers. This book discusses the conjecture of Goldbach; hypothesis of Gilbreath; decomposition of a natural number into prime factors; simple theorem of Fermat; and Lagrange's theorem. The decomposition of a prime number into the sum of two squares; quadratic residues; Mersenne numbers; solution of equations in prime numbers; and magic squares formed from prime numbers are also elaborated in this text. This publication is a good reference for students majoring in mathematics, specifically on arithmetic and geometry. |
| number theory problems with solutions: Friendly Introduction to Number Theory, a (Classic Version) Joseph Silverman, 2017-02-13 For one-semester undergraduate courses in Elementary Number Theory This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. A Friendly Introduction to Number Theory, 4th 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. |
| number theory problems with solutions: Introduction To Number Theory Richard Michael Hill, 2017-12-04 'Probably its most significant distinguishing feature is that this book is more algebraically oriented than most undergraduate number theory texts.'MAA ReviewsIntroduction to Number Theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers.The book begins with introductory material, including uniqueness of factorization of integers and polynomials. Subsequent topics explore quadratic reciprocity, Hensel's Lemma, p-adic powers series such as exp(px) and log(1+px), the Euclidean property of some quadratic rings, representation of integers as norms from quadratic rings, and Pell's equation via continued fractions.Throughout the five chapters and more than 100 exercises and solutions, readers gain the advantage of a number theory book that focuses on doing calculations. This textbook is a valuable resource for undergraduates or those with a background in university level mathematics. |
Problems in Algebraic Number Theory - University of Toronto …
Algebraic number theory—Problems, exercises, etc. I. Murty, Maruti Ram. II. Title. III. Series. QA247.E76 2004 512.7′4—dc22 2004052213 ISBN 0-387-22182-4 Printed on acid-free paper. ... ten more problems and solutions, and then typeset it into TEX. Chapters 1 to 8 arose in this fashion. Chapters 9 and 10 as well as the supplementary
IMO 2006 Shortlisted Problems - IMO official
Algebra A1. A sequence of real numbers a0,a1,a2,...is defined by the formula ai+1 = baic·haii for i≥ 0; here a0 is an arbitrary real number, baic denotes the greatest integer not exceeding ai, and haii = ai−baic. Prove that ai= ai+2 for isufficiently large. (Estonia) Solution. First note that if a0 ≥ 0, then all ai≥ 0.For ai≥ 1 we have (in view of haii <1
Exploring Number Theory via Diophantine Equations - College of …
Elementary Problems and Pell’s Equation Elliptic Curves Early Work Fermat’s Last Theorem ... solutions be rational numbers. (1570)Bombelli included translated parts in his Algebra. (1575)Holzmann (a.k.a. Xylander) attempted a completed ... in his book Number Theory, remarks that the birth of modern number theory happens on two occassions.
Number Theory Problems - Refkol
Number Theory Problems Amir Hossein Parvardi ∗ June 16, 2011 I’ve written the source of the problems beside their numbers. If you need solutions, visit AoPS Resources Page, select the competition, select the year and go to the link of the problem. All (except very few) of these problems have been posted by Orlando Doehring (orl). Contents 1 ...
Exercises Number Theory I - Olympiad
Exercises Number Theory I 1 Divisibility Beginner ... to give the problems a good try before looking up the solutions! 4.1 (Preliminary round 2012, 1.) Determine all pairs (m;n) of positive integers such that mn divides (m+1)(n+2). 4.2 (Preliminary round 2004, 1.) Find all positive integers a, b and n such that the following
Introduction to Higher Mathematics Unit #4: Number Theory
Number theory is the study of the set of positive whole numbers 1;2;3;4;5;6;7;:::; ... In this section we describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day. Sums of Squares I. Can the sum of ...
Instructor Solutions Manual Number Theory Through Inquiry
Introduction to Number Theory Solutions Manual Mathew Crawford,2008 Elementary Number Theory with Applications Thomas Koshy,2001-10 ... Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in
IMO2019 Shortlisted Problems with Solutions - IMO official
Problems (with solutions) 60th International Mathematical Olympiad Bath — UK, 11th–22nd July 2019. Note of Confidentiality ... Prove that for every real number rwith 0 ď rď 2n´ 2 you can choose a subset of the blocks whose total weight is at least rbut at most r`2. (Thailand) C3.
Number Theory Problems Solutions (book) - 220 …
Number Theory Problems Solutions ccna wireless self practice review questions for the wireless track 2015 edition with 60 questions. mini cooper 2004 This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of
NumberTheory Lecture Notes - GitHub Pages
0), all solutions are of the form x= x 0 + k·b d, y= y 0 −k·a d fork∈Z. Proof. Iftheequationhasasolution(x 0,y 0) thenobviouslyd|ax 0 +by 0 = c. Conversely,ifc= dlthensinced= am+bnforsomeintegersm,n,weknow that(ml,nl) isasolution. If (x,y) and (x 0,y 0) are two solutions then a(x−x 0) + b(y−y 0) = 0. Denote u= x−x 0, v= y 0 −y ...
(AMC 8) - Number Theory problems American Mathematics …
(AMC 8) - Number Theory problems Try these AMC 8 Number Theory Questions and check your knowledge! AMC 8, 2019, Problem 1 Ike and Mike go into a sandwich shop with a total of to spend. Sandwiches cost each and soft drinks cost each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks.
Additive Number Theory - Springer
Addictive Number Theory Melvyn B. Nathanson A True Story In 1996, just after Springer-Verlag published my books Additive Number Theory: The Classical Bases [34]andAdditive Number Theory: Inverse Problems and the Geometry of Sumsets [35], I went into my local Barnes and Noble superstore on Route 22 in Springfield, New Jersey, and looked for them on the shelves.
What Is Number Theory? - Brown University
What Is Number Theory? Number theory is the study of the set of positive whole numbers 1;2;3;4;5;6;7;:::; which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here ...
Selected Number Theory Exercises - University of Notre Dame
Exercise 20 Let kbe a natural number. (a) Using two di erent methods, ind a natural number nsuch that no natural num-ber less than kand greater than 1 divides n. (b) Using two di erent methods, nd a natural number nsuch that every natural number less than kdivides n. Exercise 21 Suppose m;n>0. Prove that xn 1 divides xm 1 if and only if n ...
4 Solutions to Introductory Problems - Springer
92 104 Number Theory Problems Solution: By division we find that n3 + 100 = (n + 10)(n2 − 10n + 100)−900.Thus, if n +10 divides n3 +100, then it must also divide 900. Moreover, since n is maximized whenever n +10 is, and since the largest divisor of 900 is 900, we must have n +10 = 900. Therefore, n = 890. 4. Those irreducible fractions! (1) Let n be an integer greater than 2.
Lecture 4: Number Theory - Harvard University
Lecture 4: Number Theory Number theory studies the structure of integers and equations with integer solutions. Gauss called it the ”Queen of Mathematics”. In this lecture, we look at a few theorems and open problems. An integer larger than 1 which is divisible only by 1 and itself is called a prime number. The number 243112609 − 1 is the ...
ANALYTIC NUMBER THEORY NOTES - Harvard University
ANALYTIC NUMBER THEORY NOTES 7 we obtain Z 1 0 A (a)2A ( 2a)da counts the number of triples (x,y,z) with x +z = 2y. This includes jA jtrivial solutions, so we want to see this integral is larger. We might expect d3N2 solutions. But now, it’s a bit hard to see how to actually bound this integral. Exercise 2.12 (Vague exercise). If, “away ...
complex - University of Oxford
3The term ‘complex number’ is due to the German mathematician Carl Gauss (1777-1855). Gauss is considered by many the greatest mathematician ever. He made major contributions to almost every area of mathematics from number theory and non-Euclidean geometry, to astronomy and magnetism. His name precedes a multitude of theorems and definitions
MATH 324 Summer 2011 Elementary Number Theory Solutions …
Elementary Number Theory Solutions to Practice Problems for Final Examination Thursday August 11, 2011 Question 1. Find integers x and y such that 314x+159y = 1: Solution: Using the Euclidean algorithm to nd the greatest common divisor of a = 314 and b = 159; we have 314 = 1 159+155 159 = 1 155+4 155 = 38 4+3 4 = 1 3+1 3 = 3 1+0
Regional Mathematical Olympiad-2019 problems and solutions
Regional Mathematical Olympiad-2019 problems and solutions 1.Suppose xis a nonzero real number such that both x5 and 20x+ 19 x are rational numbers. Prove that xis a rational number. Solution:Since x 5is rational, we see that (20x) and (x=19)5 are rational numbers. But (20x)5 19 x 5 = 20x 19 x (20x)4 + (203 19)x2 + 202 192 + (20 193) 1 x2 + 194 ...
Problems & Solutions - Georg Mohr
Baltic Way 2011 Problems & Solutions Number Theory Number Theory N-1 DEN Let a be any integer. De ne the sequence x 0;x 1;::: by x 0 = a, x 1 = 3 and x n = 2x n 1 4x n 2 + 3 for all n > 1: Determine the largest integer k a for which there exists a …
Number Theory - Stanford University
Number Theory ii COLLABORATORS TITLE : Number Theory ACTION NAME DATE SIGNATURE WRITTEN BY Ben Lynn 1980-01-01 REVISION HISTORY NUMBER DATE DESCRIPTION NAME. Number Theory iii ... Euclid’s Algorithm We will need this algorithm to fix our problems with division. It was originally designed to find the greatest common divisor of two …
MC25N AMC 8/MathCounts Advanced Number Theory
Number Theory Chapter 1: Gauss Sums Sums of arithmetic sequences (e.g. sum of the rst n positive integers) Sum of the rst n perfect squares, cubes Sample Problem: (Richard Spence) The sum 1 3+2 +33 +:::+n3 is equal to a perfect fourth power, where n 1. What is the smallest possible value of n? Chapter 2: Primes & Prime Factorization
3 Congruence - New York University
V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical in the art of cryptography. De nition 3.1 If a and b are integers and n>0,wewrite a b mod n to mean nj(b −a). We read this as \a is ...
Olympiad Number Theory Through Challenging Problems
number theory problems: Experiment with small cases. For example, while solving the following problem: Example 0.1.1. (2007 ISL) Let b;n > 1 be integers. For all ... 1Disclaimer: I did not solve all of the problems myself, the solutions that were reworded are sourced accordingly with numbers. 3. Olympiad Number Theory Justin Stevens Page 4 ...
Algebraic number theory LTCC 2008 Solutions to Problem Sheet 1
Algebraic number theory, Solutions to Problem Sheet 1, LTCC 2008 3 Let [A] denote the class of the (fractional) ideal A in the class groupCl(K).By Step 2 the ideal A4 is principal, thus [A]4 = [A4] is the identity element in Cl(K).It follows that the order of [A] in Cl(K) divides 4.But Step 2 shows that the ideal A2 is not principal which implies that [A] and
23rd Junior - Olympiad Geometry
Shortlisted Problems with Solutions June 20 - 25 2019, Agros, Cyprus. Note of Confidentiality The shortlisted problems should be kept strictly confidential until JBMO 2020 ... NUMBER THEORY N1. Find all prime numbers p for which there are non-negative integers x, y and z such that the number
TOPICS IN NUMBER THEORY - University of the Witwatersrand
Hints and solutions Additional problems. Introduction: induction Number theory is one of the most important and oldest branches of mathematics and is concerned with the properties of the natural numbers, or positive integers 1;2;3;4;:::. (Sometimes 0 is also regarded as a natural number, but we shall keep it apart.)
Seventh Edition - WordPress.com
2 Divisibility Theory in the Integers 13 2.1 Early Number Theory 13 2.2 The Division Algorithm 17 2.3 The Greatest Common Divisor 19 2.4 The Euclidean Algorithm 26 2.5 The Diophantine Equation ax +by = c 32 3 Primes and Their Distribution 39 3.1 The Fundamental Theorem of Arithmetic 39 3.2 The Sieve of Eratosthenes 44 3.3 The Goldbach Conjecture 50
Putnam Solutions - vraman23.github.io
the number of heads is even after ktosses. Then, the total number of heads is odd if we ip a head on the k+ 1-th toss. Similarly, if the number of heads is odd after ktosses, then the total number of heads is odd if we ip a tail on the k+ 1-th toss. Putting this together gives P(k+ 1) = (1 P(k))p k+1 + P(k)(1 p k+1) = P(k)(1 2p k+1) + p k+1 = P ...
Algebraic Number Theory - uniroma2.it
Algebraic Number Theory 1. Introduction. An important aspect of number theory is the study of so-called “Diophantine” equations. These are (usually) polynomial equations with integral coefficients. The problem is to find the integral or rational solutions. We will see, that even when the original problem involves only ordinary
Number Theory I - Solutions - Olympiad
Number Theory I - Solutions 1 Divisibility Beginner 1.1Showthat900divides10!. Solution: 900 = 25910j12345678910 = 10! ... Solution: When nis odd, let’s write n= 2k+ 1 for a natural number k 3. Then we have that n= (k+ 1) + kwhere gcd(k;k+ 1) = 1 and we’re done. When nis even, we have that n= 2kforanaturalnumberk>3. Inwhichcase ...
Formalizing IMO Problems and Solutions in Isabelle/HOL
Currently problems in algebra, combinatorics and number theory are formalized (geometry problems are skipped, since their formalization requires a rich background theory of high-school synthetic geometry, that is not available in Isabelle/HOL at the moment). All formalized solutions and problem statements
Number theory - Diophantine equations - University of Toronto ...
3.4 Problems 18.Find two nontrivial solutions to x2 8y2 = 1. 19.Let t n = 1 + 2 + 3 + :::+ n denote the nth triangular number. Find all triangular numbers which are perfect squares. 20. (Putnam 2000/A2) Prove that there exists in nitely many integers n such that n;n+ 1, and n+ 2 are each the sum of two squares of integers.
Collection of problems in probability theory - Gwern
The problems of the second section are intended for those who are primarily interested in applying the theory to statistics. In these problems we use the following notation: N is the total number of objects under consideration; N{ } is the number of these objects having the property appearing in the braces.
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Number theory - Modular Arithmetic - University of Toronto …
2.1 Problems 5.Let k = 20082 + 22008. What is the unit digit of 2k + k2? 6.Find n such that n 5= 275 + 845 + 110 + 1335. 7.Let p be a prime not equal to 2, 3 or 5. Let n be the p 1 digit number all of whose digits are 1. Show that n is divisible by p. 8.Let p > 2 be a prime number and n a positve integer. Prove that p divides 1 pn +2pn +:::+(p ...
Computations in Number Theory Using Python: A Brief Introduction
7 Problems 23 1 Introduction The aim of these notes is to give a quick introduction to Python as a language for doing computations in number theory. To get an idea of what is possible, we rst observe that Python can be used as a calculator. Imagine that we are using our portable unix laptop, logged in as student. The rst step is to type the ...
Some Unconventional Problems in Number Theory - JSTOR
Some Unconventional Problems in Number Theory A m6lange of simply posed conjectures with frustratingly elusive solutions. PAUL ERDOs Hungarian Academy of Sciences University of Colorado Boulder, CO 80309 I state some curious, unusual, and mostly unsolved problems in various branches of number theory. Factorial Powers 1. Put f(n) = E(l /p) for p ...
Module MA3411: Galois Theory Worked Solutions to Problems Michaelmas ...
3. Let dbe a rational number that is not the square of any rational number, let p dbe a complex number satisfying (p d)2 = d, and let Ldenote the set of all complex numbers that are of the form a+ b p dfor some rational numbers aand b. Prove that Lis a sub eld of the eld of complex numbers, and that L:Q is a nite eld extension of degree 2. If z ...
Problems in Algebraic Number Theory
II Solutions 137 1 Elementary Number Theory 139 1.1 Integers 139 1.2 Applications of Unique Factorization 146 1.3 The ABC Conjecture 150 1.4 Supplementary Problems 153 2 Euclidean Rings 159 2.1 Preliminaries 159 2.2 Gaussian Integers 161 2.3 Eisenstein Integers 165 2.4 Some Further Examples 167 2.5 Supplementary Problems 169
Number Theory Cheat Sheet Edexcel FP2 - Physics & Maths Tutor
Write the general form of a three-digit number using the form above 𝑁𝑁= 100(𝑎𝑎) + 10(𝑏𝑏) + 𝑐𝑐 𝑎𝑎 ,𝑏𝑏𝑐𝑐∈{0,1,2,3,4,5,6,7,8,9} 𝑎𝑎≠0 Use the modulo of 100 and 10 to rewrite the number in terms of the decimal digits 100 ≡1 mod 9 10 ≡1 mod 9 For the number to be divisible by 9, it must equal
Analytic number theory : exploring the anatomy of integers - GBV
vi Contents §9.8. TheDickmanfunction 147 §9.9. Consecutive smoothnumbers 149 Problems onChapter 9 150 Chapter 10. TheHardy-RamanujanandLandauTheorems 157 §10.1. TheHardy-Ramanujaninequality 157 §10.2. Landau'stheorem 159 ProblemsonChapter 10 164 Chapter 11. Theabc Conjectureand SomeofIts Applications 167 §11.1. The abc conjecture 167 §11.2. …
Methods of Solving Number Theory Problems - GBV
5.2 Answers and Solutions to the Homework 340 References 377 Appendix 1. HistoricOverview ofNumberTheory 381 Appendix 2. MainDirections inModernNumberTheory 385 Index 389. Title: Methods of Solving Number Theory Problems Subject: Cham, Springer International Publishing, 2018 Keywords: Signatur des Originals (Print): T 18 B 3322. Digitalisiert ...
Preface - AwesomeMath
Therefore, 121 Number Theory Problems for Mathe-matics Competitions is a new book aimed at those who have little or moderate ... The book contains many examples, proposed problems followed by solutions, and an ample selection from many recent mathematical contests and Olympiads, journals, as well as tests given to the students during
Elementary Number Theory - 2nd Ed. - isinj.com
problems and should not feel discouraged if some are baffling. There is benefit in trying to solve prOblems whether a solution is found or not. I. A. Barnett has written [1] "To discover mathematical talent, there is no better course in elementary mathematics than number theory. Any student who can work the exercises in a modern text in number ...
Number Theory Problems Solutions - appleid.ultfone.com
27 Sep 2024 · 2 Number Theory Problems Solutions 2021-02-08 historical approach which mainly concerns us is the determination of the problems which suggested the theorems, and the study of which provided the concepts and the techniques which were later used in their proof. In most number theory books residue classes are introduced prior to
PROBLEMS IN ELEMENTARY NUMBER THEORY - tomlr.free.fr
PROBLEMS IN ELEMENTARY NUMBER THEORY Hojoo Lee, Version 0.77 [2003/11/24] God does arithmetic. Gauss Contents 1. Preface 2 2. Notations and Abbreviations 3 3. Divisibility Theory I 4 4. Divisibility Theory II 8 5. Arithmetic in Zn 11 6. Primes and Composite Numbers 13 7. Rational and Irrational Numbers 15
Putnam Practice: Number theory - Department of Mathematics
2 (8) Showthatforeverynthesequence2,22,2 22,2 2 2,:::(mod n)iseventuallyconstant. Herearetwomore“themed” problems, involving Pell’s equation: if d ...
