Theorem if x and y are rational numbers, then xy is a rational number. 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. The problems are culled from various sources, many from actual contests and olympiads, and in general are very di. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures. Some numbertheoretic problems that are yet unsolved are. Olympiad number theory through challenging problems. Multiplicative algebraic number theory is the dominant overarching theme.
A repeated product can be written using product notation n 1 o k0 a k a0a1 an 1 awrite n factorial n. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Cse 1400 applied discrete mathematics number theory and proofs. Number theory is one of the oldest and most beautiful branches of mathematics. Prime numbers number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles leedham. The following theorems illustrate a number of important properties of divisibility. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. 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. 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. Jul 11, 2007 chapter 1 introduction the heart of mathematics is its problems.
A lively introduction with proofs, applications, and stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. Analytic and modern tools, henri cohen, springerverlag graduate texts in mathematics 239 and 240, may 2007, errata pdf equidistribution in number theory, an introduction, proceedings of the nato advanced study institute on equidistribution in number theory, montreal, canada, 11. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics.
Proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe. Divisibility the notion of divisibility is the central concept of one of the most beautiful subjects in advanced mathematics. Number theory has always fascinated amateurs as well as professional mathematicians. Number theory, branch of mathematics concerned with properties of the positive integers 1, 2, 3. Other proofs in the early 20th century mostly used tauberian theorems, as in wiener 1932, to extract the prime number theorem from the nonvanishing of s on res 1. Cse 1400 applied discrete mathematics number theory and proofs department of computer sciences college of engineering florida tech spring 2012 problems labeled. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Number theory a lively introduction with proofs applications. 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. Eulers theorem is a generalization of fermats little theorem and the proof presented here has the same form as the proof. 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.
Despite their ubiquity and apparent simplicity, the natural integers are chockfull of beautiful ideas and open problems. He proved the fundamental theorems of abelian class. Introduction to number theory lecture notes adam boocher 20145, edited by andrew ranicki 20156 december 4, 2015 1 introduction 21. Some of his famous problems were on number theory, and have also been in. In this chapter, we will explore divisibility, the building block of number theory. Riemanns explicit formula already in the number theory of z, the relationship between primes and analytic properties of the riemanneuler s is striking. Sometimes called higher arithmetic, it is among the oldest and most natural of mathematical pursuits. Properties of integers proving existential statements universal statements and basic techniques of direct proof common mistakes getting proofs started disproof by counterexample proof by contradiction. The first eleven such numbers are 2, 3, 5, 7, 11, 17, 19, 23, 29, and 31. 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. If you have the right audience for it, you will enjoy teaching from this book as well. Karl friedrich gauss csi2101 discrete structures winter 2010. Proofs are given when appropriate, or when they illustrate some insight or important idea.
A natural number p is called a prime number if it has exactly two distinct natural number divisors, itself and 1. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The idea of proof is central to all branches of mathematics. Branislav kisaeanin trations of abstract mathematical concepts and to learn a bit about their historic. Goldbachs conjecture is every even integer greater than 2 the sum of distinct primes. Of course, even if the subject is proof itself, we need to prove something, so in this chapter we begin our study of number theory, that is, the properties of the integers often, but not always, the nonnegative integers. A friendly introduction to number theory is an introductory undergraduate text designed to entice nonmath majors into learning some mathematics, while at the same time teaching them how to think mathematically. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course.
The proof of independence of ch by forcing is added. A number of computer languages have builtin functions that enable you to compute many values of q and r for the quotientremainder theorem. Combinatorics, number theory, and 3 treats number theory, and chapter 4 focuses on geometry. Number theory a lively introduction with proofs, applications, and stories pdf. I hadnt realized that i hadnt written it in the notes. Paul halmos number theory is a beautiful branch of mathematics. For example, here are some problems in number theory that remain unsolved.
A prime number is an integer greater than 1 whose only positive divisors are itself and 1. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory. The formal prerequisites for the material are minimal. Pages in category theorems in number theory the following 109 pages are in this category, out of 109 total. Erdos 1950 and selberg 1950 gave proofs of the prime number theorem elementary in the sense of using no complex analysis or other limiting procedure devices. It abounds in problems that yet simple to state, are very hard to solve. A primary focus of number theory is the study of prime numbers, which can be. Letting pn denote the number of primes p b n, gauss conjectured in the early. If a and b are integers and there is some integer c such that a bc, then we say that b divides a or is a factor or divisor of a and write ba.
In this section we will describe a few typical number theoretic problems. This chapter will introduce many important concepts that will be used throughout the rest of the book. There are plenty of exercises, and the book is a pleasure to read. Number theoryelementary divisibility wikibooks, open books. There is an indication of a considerable simpli cation of my forcing treatment, by restricting the construction of.
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