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. Find materials for this course in the pages linked along the left. There are as many kinds of average as there are purposes. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. Some numbertheoretic problems that are yet unsolved are. A good one sentence answer is that number theory is the study of the integers, i. It abounds in problems that yet simple to state, are very hard to solve. One of the central goals of this course is also to introduce you to mathematical proof and the way in which professional mathematics is done. Poonens course on algebraic number theory, given at mit in fall 2014. This chapter will introduce many important concepts that will be used throughout the rest of the book. To receive announcements by email, add yourself to the nt mailing list. Basically, it tells you how fast a function grows or declines.
For many of us, a number is just a number, a bit of information that tells you, say, what time it is. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. Elementary number theory jim belk january 27, 2009 number theory is the branch of mathematics concerned with the properties of the positive integers, such as divisibility, prime numbers, and so forth. With more than 2,200 courses available, ocw is delivering on the promise of open sharing of knowledge. Applications cse235 introduction hash functions pseudorandom numbers representation of integers euclids algorithm c.
Big o notation mit massachusetts institute of technology. This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. In this article we shall look at some elementary results in number theory, partly because they are interesting in themselves, partly because they are useful in other contexts for example in olympiad problems, and partly because they will give you a flavour of what number theory is about. The resolution of fermats last theorem by wiles in 1995 touched off a flurry of related activity that continues unabated to the present, such as the recent solution by khare and wintenberger of serres conjecture on the relationship between. 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. Complete lecture notes and the following lecture notes courtesy of oron propp. However, other fixed basket approaches are also possible. The resolution of fermats last theorem by wiles in 1995 touched off a flurry of related activity that continues unabated to the present, such as the recent solution by khare and wintenberger of serres conjecture on the relationship between mod p galois. Cryptography hash functions iii however, the function is clearly not onetoone. This course is an introduction to analytic number theory, including the use of zeta functions and lfunctions to prove distribution results concerning prime numbers e. On the website you can also help me collecting solutions for the problems in the book all available solutions will be on the website only. Elementary number theory lecture notes lior silberman.
But mathematicians look at that same number and divine relationships that underlie nature itself. With more than 2,400 courses available, ocw is delivering on the promise of open sharing of knowledge. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, in addition to a careful presentation of the standard material usually taught in a first course in elementary number. I have made them public in the hope that they might be useful to others, but these are not o cial notes in any way. The csec mathematics syllabus objective for this section are students should. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in chapters 2 and 3. 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. Analytic number theory mathematics mit opencourseware. Then we will quickly progress to congruences, fermats little theorem not his last and the number theory behind rsa encryption which is central to internet security.
Readings and lecture notes topics in algebraic number theory. To find all of the prime numbers between 1 and 100, complete the following exercise. The majority of students who take courses in number theory are mathematics majors who will not become number theorists. 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. It is easy to see that a nonempty subset sof zis a subgroup of zif and only if x y2sfor all x2sand y2s. At its core, number theory is the study of the integer ring z. This pdf document contains hyperlinks, and one may navigate through it by click.
A computational introduction to number theory and algebra version 2 victor shoup. Olympiad number theory through challenging problems. Mit department of mathematics number theory seminar. Some typical number theoretic questions the main goal of number theory is to discover interesting and unexpected relationships between different sorts of numbers and to prove that these relationships are true. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles leedham. Assignments number theory i mathematics mit opencourseware.
This is a list of number theory topics, by wikipedia page. Notations and conventions standard notations for sets z ring of all integers n 0 set of all integers. If we compute mod n on both sizes, nqc cancels out and we obtain ac bc mod n. Number theory authorstitles recent submissions arxiv. The integers and prime numbers have fascinated people since ancient times. In this article we shall look at some elementary results in number theory, partly because they are interesting in themselves, partly because they are useful in other contexts for example in olympiad problems, and partly because they will give you a. A computational introduction to number theory and algebra. Basic algorithms in number theory universiteit leiden. In this chapter, we will explore divisibility, the building block of number theory. When two elements, x 1 6 x 2 hash to the same value, we call it a collision. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, in addition to a careful presentation of the standard material usually taught in a first course in elementary.
Goldbachs conjecture is every even integer greater than 2 the sum of distinct primes. Number theory is one of the oldest and most beautiful branches of mathematics. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry. Mit department of mathematics number theory seminar partition statistics, quasimock theta functions, and weak maass forms karl mahlburg mit abstract. Topics covered include primes, congruences, quadratic reciprocity. Practice number theory problems 3 b show that if a b mod n, then for all positive integers c, ac bc mod n. In this lecture we introduce the notion of a local field. One such result concerns the halting problem, which asks for an algorithm that takes as input a computer program pand an integer x, and out. Karl friedrich gauss csi2101 discrete structures winter 2010. 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. Number theory is the study of the set of positive whole numbers. What are the \objects of number theory analogous to the above description. Big o notation with a capital letter o, not a zero, also called landaus symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions.
In this section we will describe a few typical number theoretic problems. This course is an elementary introduction to number theory with no algebraic prerequisites. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. 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. The websites by chris caldwell 2 and by eric weisstein are especially good. The formal prerequisites for the material are minimal. It covers the basic background material that an imo student should be familiar. 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. Lectures on analytic number theory tata institute of.
By the fundamental theorem of arithmetic, every element of z can be. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Mit opencourseware makes the materials used in the teaching of almost all of mit s subjects available on the web, free of charge. Lecture notes analytic number theory mathematics mit.
Instead of choosing the basket of period 0 or 1 or an average of these two baskets, it is possible to choose a basket that pertains to an entirely different period, say period b. Lecture notes 3 iv before we start our proof, we want to point out that this statement is a generalization of the previous one. Introduction the answer to the question what is the mean of a given set of magnitudes cannot in general be found, unless there is given also the object for the sake of which a mean value is required. Solutions to problem sets were posted on an internal website. For titles and abstracts, please see the mathematics department calendar. We focus ourselves to some classical results on the prime divisor counting function.
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