The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. This is a list of number theory topics, by wikipedia page. Possible topics for phd theses in mathematics department. Number theorist leonard dickson once said, after all, thank god that number theory is unsullied by any application. Photograph your local culture, help wikipedia and win. Map pdf a printable pdf file depicting the original 1997 map created by dr. Topics of further study include attempts to discover if there is an infinite amount of pairs, to discern patterns, and to better understand why and how this happens. Theons work contains more topics and was a far superior work mathema. Feb 01, 2002 number theory, the branch of mathematics that studies the properties of the integers, is a repository of interesting and quite varied problems, sometimes impossibly difficult ones. Though we now understand that number theory has boundless applications, uses, and purposes, it can appear to be frivolous to the point of pointlessness especially the subset known as recreational number theory. Need to get past finetuning need to see radical change revenue sources are challenged and will. That said, you can still do interesting problems and write them up. It also includes an introduction to discrete probability theorythis material is needed to properly treat the topics of probabilistic algorithms and cryptographic applications.
Massachusetts institute of technology department of mathematics. Number theory is one of the oldest and most beautiful branches of mathematics. Since this is an introductory textbook, and not an encyclopedic reference for specialists, some topics simply could not be covered. A masterclass, offers a more comprehensive approach, adding additional material in further chapters and appendices. Topics in number theory is essentially a first course in number theory and as a prerequisite requires familiarity not much more than what is covered in any high school mathematics curriculum. Since his paper is written for a manual of physics, he does not. For example, if n assumes the value 2, 3, 5, or 7, the expression 2 n.
A2a as a high schooler, you likely dont have the prerequisites to do actual mathematical research. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. By undertaking a project with us, youll have the chance to create change within a range of diverse areas. You can see a list of our 10 hot leadership topics for 2020 right here. This function has been extensively studied, and is still the object of current research. By writing the preface, i am afforded the opportunity to express my gratitude to ken for beng the inspiring and driving force behind the whole conference. All related to properties of lattices in e n and bases, the dense ball packing problem, the minkowskihlawka theorem, etc, and can range to reduction of polynomials or coding. The treatment of all these topics is more or less standard, except that the text only deals with commutative structures i. 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. Topics to be covered include number fields, class numbers, dirichlets units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. Pdf in this paper i draw upon a few images of number theory as a queen and as a servant of mathematics. From july 31 through august 3,1997, the pennsylvania state university hosted the topics in number theory conference.
The ramification theory needed to understand the properties of conductors from the point of view of the herbrand distribution is given in c. Trends are emerging now that help reveal how the future of work is shaping up for 2020 and beyond. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. Thank god that number theory is unsullied by any application. 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. Find materials for this course in the pages linked along the left. What are the \objects of number theory analogous to the above description. This book provides careful coverage of all core topics in a standard introductory number theory course with pointers to some exciting further directions.
An example is checking whether universal product codes upc or international standard book number isbn codes are legiti mate. Mathematics archives topics in mathematics number theory. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms. Basic algorithms in number theory universiteit leiden. A masterclass presents a fresh take on congruences, power residues, quadratic residues, primes, and diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of pascals triangle mod p, fermats last theorem for polynomials, and. An additional theme running throughout the course will be the use of computer algebra to investigate.
A good one sentence answer is that number theory is the study of the integers, i. Olympiad number theory through challenging problems. Computational number theory is also known as algorithmic number theory. Comments suggestions, criticisms, and comments of all types are appreciated as they serve to better inform our efforts to improve this site. Then mz the set of integer multiples of m is a subgroup of z. A subset s of the set z of integers is a subgroup of z if 0.
Msri model theory, arithmetic geometry and number theory. Here are nevertheless some suggestions for possible research topics, which the department is particularly qualified to supervise. Number theory, branch of mathematics concerned with properties of the positive integers 1, 2, 3. Sometimes called higher arithmetic, it is among the oldest and most natural of mathematical pursuits.
Contributors this site is the result of the collobaration of a number of individuals. Whereas one of the principal concerns of the latter theory is the deconposition of numbers into prime factors, additive number theory deals with the decomposition of numbers into summands. Find all the books, read about the author, and more. Number theory, software sites, computational number theory sites, conferences, courses in number theory, lecture notes, journals on a generalized fermatwiles equation add. Topics in elementary number theory before start studying of cryptography, here is some background that shall be required. It abounds in problems that yet simple to state, are very hard to solve. Divisibility, the fundamental theorem of number theory 1 1. Im not so sure if every programmer should know some number theory knowledge. In particular, we read about combinatorial problems in number theory, a branch of. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate.
Given a natural number n, is it prime or composite. Basic algorithms in number theory 27 the size of an integer x is o. It is intended to be an introduction to the theory of nu. Part of the the university series in mathematics book series usma.
By writing the preface, i am afforded the opportunity to express my gratitude to ken for beng the inspiring and driving force. It is easy to see that a nonempty subset sof zis a subgroup of zif and only if x y2sfor all x2sand y2s. The websites by chris caldwell 2 and by eric weisstein are especially good. In this section we will meet some of the concerns of number theory, and have a brief revision of some of the relevant material from introduction to algebra. Number theory has always fascinated amateurs as well as professional mathematicians. Im not that well versed in number theory, but since you mention computer sc.
The conference was organized by ken ono and myself. Number theory is the study of the set of positive whole numbers. Number theory, the branch of mathematics that studies the properties of the integers, is a repository of interesting and quite varied problems, sometimes impossibly difficult ones. Higher education principles higher education provides equal opportunity for all equal access equal chance of succeeding must deliver value high quality, affordable and access and be able to measure it.
The choice of topics covered in this book was motivated primarily by their. Many of the students have an undergraduate course in abstract algebra and a course in real analysis but few have any complex analysis background. Number theory level 4 for a positive integer n n n, let. Euclid proved that a number n of the form 2 n12 n1 is a perfect number if the factor 2 n1 is prime. Students are also introduced to beautiful enlightening questions like the structure of pascals triangle mod p, fermats last theorem for polynomials, and modern twists on. Topics for an undergraduate expository paper in number theory. Introduction to number theory description integers, divisibility, prime numbers, unique factorization, congruences, quadratic reciprocity, diophantine equations and arithmetic functions. For example, if n assumes the value 2, 3, 5, or 7, the expression 2 n 1 takes on the value 3, 7, 31, or 127, all of which are prime. I assume you are asking for mustknow knowledge for algorithm programming contests e. We shall consider the corresponding questions for the distribution of p4 modulo 1.
A computational introduction to number theory and algebra. It is now independent, with many applications and connections. Lectures on analytic number theory tata institute of. For phd studies in mathematics, it is not absolutely necessary to choose a phd topic at the time of the application. Number theory web american site and australian site add. The definitions and elementary properties of the absolute weil group of a number field given in chapter ii, 2. Citescore values are based on citation counts in a given year e. Prerequisites one variable calculus topics the integers divisibility prime numbers greatest common divisor euclidean algorithm. Algebra and number theory school of mathematics and. Topics in the theory of numbers undergraduate texts in. 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.
What are some ideas for a research projectpaper on number. These notes were prepared by joseph lee, a student in the class, in collaboration with prof. In particular, if we are interested in complexity only up to a. It will be useful to extend the theory of formal power series to allow us to. Elementary number theory usf department of mathematics. A computational introduction to number theory and algebra version 2 victor shoup. Some numbertheoretic problems that are yet unsolved are.
In this book, the authors have gathered together a collection of problems from various topics in number theory that they find beautiful, intriguing, and from a. An introduction presents a fresh take on congruences, power residues, quadratic residues, primes, and diophantine equations, as well as hot topics like cryptography, factoring, and primality testing. The mission of the journal is to publish highquality original articles that make a significant contribution to these research areas. What are the most critical issues for leaders to watch out for in the coming year, and which will have the most substantial impact on the workplace as we head into the. A phd student can freely choose the subject for herhis thesis. Factoring out a root of a 64 polynomial, modulo to. This course is a first course in algebraic number theory. 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. 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. List of recreational number theory topics topics in cryptography. Estimates of some functions on primes and stirlings formula 15 part 1. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the.