Elementary number theory primes, congruences, and secrets. These notes serve as course notes for an undergraduate course in number the ory. In this book, all numbers are integers, unless specified otherwise. Sample student work all student work is used with permission. From zero to infinity, an entertaining list of every number that counts. Life is full of patterns, but often times, we do not realize as much as we. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. Projects related resources download course materials. Topics in number theory, volumes i and ii dover books on mathematics kindle edition by leveque, william j download it once and read it on your kindle device, pc, phones or tablets. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington.
An introduction to the theory of numbers, 6th edition, by g. Topics in number theory by leveque, william judson. Readings provide both practical knowledge and a theoretical framework, while assignments offer a. Five of the projects will have three participants and the sixth will have only two. The libretexts libraries are powered by mindtouch and are supported by the department of education open textbook pilot project, the uc davis office of the provost, the uc davis library, the california state university affordable learning solutions program, and merlot. What are some ideas for a research projectpaper on number. It is easy to see that a nonempty subset sof zis a subgroup of zif and only if x y2sfor all x2sand y2s. Topics in elementary number theory before start studying of cryptography, here is some background that shall be required. Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. You are free to choose which problems in this project that you would like to work on, all i ask is that you work on things which you havent seen before, and that you do not use any books or the internet. Number theory, branch of mathematics concerned with properties of the positive integers 1, 2, 3. The orange ball marks our current location in the course.
Im not that well versed in number theory, but since you mention computer sc. Feb 01, 2002 this rather unique book is a guided tour through number theory. A2a as a high schooler, you likely dont have the prerequisites to do actual mathematical research. Msc course content in classes is imparted through various means such as lectures, projects, workshops m.
My goal in writing this book was to provide an introduction to number theory and algebra, with an emphasis. Algebra and number theory school of mathematics and. The recommended books are 1 h davenport, the higher arithmetic. A computational introduction to number theory and algebra. The chapter on the work of chowla and pillai is part of an article written in collaboration with r thangadurai that appeared in resonance. Thus, for example, in chapter 1, sections 1 through 3 form a topic, as do sections 4 and 5. Topics for an undergraduate expository paper in number theory. Number theory has always fascinated amateurs as well as professional mathematicians. While most introductions to number theory provide a systematic and exhaustive treatment of the subject, the authors have chosen instead to illustrate the many varied subjects by associating recent discoveries, interesting method, and unsolved problems. The point of these notes is not in t he rst place to serve as. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. While most introductions to number theory provide a systematic and exhaustive treatment of the subject, the authors have chosen instead to illustrate the many varied subjects by associating recent discoveries, interesting methods, and unsolved problems. One of the unique characteristics of these notes is the careful choice of topics and its importance in the theory of numbers.
Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. Elementary number theory projects there are going to be. Are people still interested in this type of mathematics today. Find materials for this course in the pages linked along the left. For an explanation of the background pattern, skip ahead to the end of the page. If he had not agreed to be a coauthor, this book would have.
American libraries canadian libraries universal library community texts project gutenberg biodiversity heritage library childrens library. The formal prerequisites for the material are minimal. Topics in number theory summary mathematics libretexts. Introduction to analytic number theory fall 2015 if you find a mistake, omission, etc. High school mathematics, familiarity with proofs by mathematical induction and with the basic properties of limits of sequences of real numbers in particular the fact. It represents the parts of analytic number theory that were of greatest interest to him. Topics in number theory, volumes i and ii dover books on. It is left, as a general recommendation to the reader, to follow up any topic in further detail by reading whathac has to say. This book is designed for use in a first course in number theory at the junior or senior level. The freedom is given in the last two chapters because of the advanced nature of the topics that are presented.
Life is full of patterns, but often times, we do not realize as much as we should that mathematics too is full of patterns. List of free project topics, ideas, subjects and final year research materials. Use features like bookmarks, note taking and highlighting while reading topics in number theory, volumes i and ii dover books on mathematics. The editors are grateful to springerverlag for their helpfulness and courtesy. Section 1 on cryptography is on an application of number theory in the field of message decoding, while the other sections on elliptic curves and the riemann zeta function are deeply connected with number theory. Preface these are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. Problemsolving and selected topics in number theory.
Ribenboim, the new book of prime number records, springer. Each project will consist of an original treatment of important numbertheoretic material. In this undergraduate level seminar for mathematics majors, students present and discuss subject matter taken from current journals or books. Analytic number theory lecture notes based on davenports book andreas str ombergsson these lecture notes follow to a large extent davenports book 15, b ut with things reordered and often expanded. The systematic study of number theory was initiated around 300b. Elementary number theory projects there are going to be six di erent projects.
Algebraic number theory involves using techniques from mostly commutative algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects e. This chapter discusses various topics that are of profound interest in number theory. This book is written for the student in mathematics. There are great books on the abstract properties of prime numbers. Reallife math project cant find what youre looking for. The main objects that we study in this book are number. Find, read and cite all the research you need on researchgate.
In this volume, some of the expositions related to the former topic have been put together. Project interactivate activities number and operation concepts. Wright article pdf available in contemporary physics 5. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the. Sometimes called higher arithmetic, it is among the oldest and most natural of mathematical pursuits. Topics in the theory of numbers undergraduate texts in. Pdf two topics in number theory a formula for primes and an. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. It is possible to use game theory to give a complete analysis of. Akash sujith favorite favorite favorite march 11, 2015 subject. This book is suitable as a text in an undergraduate number theory. Scholastic teachables worksheets, lesson plans, learning games, and more.
Many areas of modern mathematics such as number theory, homological algebra, algebraic geometric, etc. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. In each of the three visual mathematics courses, students explore topics including geometry, measurement, algebra, probability, statistics, numeration, number theory, operations, and estimation. There are no references made to other work in this book. By undertaking a project with us, youll have the chance to create change within a range of diverse areas. It is useful in many branches of mathematics, including number theory and applied mathematics. Two topics in number theory a formula for primes and an algebraic identity. Each project will consist of an original treatment of important number theoretic material. Mathematics archives topics in mathematics number theory. Many of the students have an undergraduate course in abstract algebra and a course in real analysis but few have any complex analysis background. Our textbook contains a little material on each project. Project approach study guide this study guide offers educators an overview of the project approach and guides them through the process of developing and implementing a project in the classroom. The author has certainly planned his book well, chosen material that will be stimulating to its intended audience, and carried the project through in such a way that interest seldom.
In this section we will describe a few typical number. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. Its exposition reflects the most recent scholarship in mathematics and its history. This is a list of number theory topics, by wikipedia page. Residue number system cunningham project quadratic residuosity problem. For example, here are some problems in number theory that remain unsolved. Students were asked to choose a serious topic in number theory of which the projects were my suggestions with useful references to write a final paper of at least 10 pages explaining their topic i. The websites by chris caldwell 2 and by eric weisstein are especially good. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles leedham. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. Get a strong understanding of the very basic of number theory. The journal of number theory features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas.
German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. This rather unique book is a guided tour through number theory. The third book in the series, number theory and combinatorics, is by. List of interesting mathematics research project topics mathematics is the abstract science of number, quantity, and space, either as abstract concepts pure mathematics or as applied to other disciplines such as physics and engineering applied mathematics. You can read the complete proof on one of my favourite books on trnascendental number theory, called making transcendence transparent. To see what is going on at the frontier of the subject, you may take a look at some recent issues of the journal of number theory which you will. That said, you can still do interesting problems and write them up. Number theory problems in mathematical competitions 2015 2016 login expand child menu. However, it is certainly challenging to present real number theory to high school students in a way which will be enjoyable and meaningful for them. Bibliography of books on history of arithmetic and number theory add. Laboratories for their support for this project, and for the opportunity to use. These are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. An illustrated theory of numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. While most introductions to number theory provide a systematic and.
It covers the basic background material that an imo student should be familiar with. I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper 10 pages. Other topics in number theory mathematics libretexts. Chapter 30 of your book discusses fermats theorem for n 4, and chapter 46 discusses. Number theory, known to gauss as arithmetic, studies the properties of the integers. The course of masters of science msc postgraduate level program offered in a majority of colleges and universities in india. Hamilton cycle decompositions of cayley graphs and related topics. Introduction to number theory and its applications lucia moura winter 2010 \mathematics is the queen of sciences and the theory of numbers is the queen of mathematics. Thus, the book may be useful for reference or selfstudy by readers who want to learn about cryptography, or it could also be used as a textbook in a graduate or upperdivision undergraduate course on computational number theory and algebra, perhaps geared towards computer science students. History, crash course in number theory, properties of mod, calculator algorithms, simple cryptosystems, modern stream ciphers, running time of algorithms, public. Topics in mathematics number theory mathematics archives.
While most introductions to number theory provide a systematic and exhaustive treatment of the subject, the authors have chosen instead to illustrate the many varied subjects by associating recent discoveries, interesting methods, and. Commutative algebra is the area of mathematics that studies commutative rings and other related topics such as module theory. The main objects that we study in this book are number elds, rings of integers of number elds, unit groups, ideal class groups, norms, traces, discriminants, prime ideals, hilbert and other class elds and associated reciprocity laws, zeta and l functions, and algorithms for computing with each of the above. Of course, we speak of number theory and computer experiment. Preface to the second edition v preface to the first edition vii preface to the english translation ix facts used without proof in the book xvii chapter 1. Karl friedrich gauss csi2101 discrete structures winter 2010.
Number theory and algebra play an increasingly signi. Download undergraduate projects topics and materials accounting, economics, education. The author has certainly planned his book well, chosen material that will be stimulating to its intended audience, and carried the project through in such a way that interest seldom flags. Number theory and combinatorics indian academy of sciences. Mathematics undergraduate project topics, research works and materials, largest undergraduate projects repository, research works and materials. Rademacher started work on the present volume no later than 1944. Mathematics research project topics mathematics is the abstract science of number, quantity, and space, either as abstract concepts pure mathematics or as applied to other. Number theory is right for this in part because of its accessibility. This is a textbook about classical elementary number theory and elliptic curves. Each of the 45 lessons is intended for a few days of class time.
I will also post a pdf of the two documents to the website. Visual mathematics, secondary the math learning center. In this section we will describe a few typical number theoretic problems. Finally, i would like to thank amir to join me in this project. Undergraduate project topics and masters dissertation. For the latest updates, follow us on facebook more details.
141 64 1282 449 561 554 1412 350 72 1524 816 219 451 359 305 341 89 519 1164 156 81 892 182 1200 79 1356 633 645 658 1164 148 226 857 717 817 742 87 790 806 676 718 62 1355 1229 196