Since ancient times, people have separated the natural numbers into a variety of different types. However, i wanted to discuss logic and proofs together, and found that doing both. Letting pn denote the number of primes p b n, gauss conjectured in the early. 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. Proofs are given when appropriate, or when they illustrate some insight or important idea. The systematic study of number theory was initiated around 300b. Given all statements in your theory, logically derive new statements which are true and call them propositions or theorems. The second and the third proof use special wellknown number sequences.
Number theory wikibooks, open books for an open world. This book covers an elementary introduction to number theory, with an emphasis on presenting and proving a large number of theorems. The first eleven such numbers are 2, 3, 5, 7, 11, 17, 19, 23, 29, and 31. These lectures have been compiled from a variety of sources, mainly from the recommended books. The third book in the series, number theory and combinatorics, is by. Andrews, evan pugh professor of mathematics at pennsylvania state university, author of the wellestablished text number theory first published by saunders in 1971 and reprinted by dover in 1994, has led an active career discovering fascinating phenomena in his chosen field number theory. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the. Complex numbers complex numbers of the form iy, where y is a nonzero real number, are called imaginary numbers. He laid the modern foundations of algebraic number theory by. 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. Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic. Number theory for beginners by andre weil is the slickest,most concise yet best written introduction to number theory ive ever seenits withstood the test of time very well.
Followed by an introduction to analytic number theory. Number theory a lively introduction with proofs, applications, and stories pdf. Chapter 1 covers this theory at a fairly rapid pace. The proof of independence of ch by forcing is added. These are the notes of the course mth6128, number theory, which i taught at. The primary goals of the text are to help students. He should receive credit for some of the proofs that appear in this book, for example, the new padic proof of the theorem of kummer on the second factor in the.
Add the postulates to a list and call that list theory. No attempts will be made to derive number theory from set theory and no knowledge of calculus will be assumed. Contents preface ix introduction x i fundamentals 1. We will especially want to study the relationships between different sorts of numbers. Number theoryelementary divisibility wikibooks, open books. Find materials for this course in the pages linked along the left. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Induction is covered at the end of the chapter on sequences. A natural number p is called a prime number if it has exactly two distinct natural number divisors, itself and 1. 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. It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit fivefold symmetry.
Number theory is the study of the set of positive whole numbers. For math students that have never learned number theory and want to learn it quickly and actively, this is still your best choice. 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. Probability theory is the most directly relevant mathematical background, and it is assumed that the reader has a working knowledge of measuretheorybased probability theory. Addressing the importance of constructing and understanding mathematical proofs, fundamentals of mathematics.
Bookofproofs mathematics, physics and computer science. Number theory is the study of the set of positive whole numbers 1. Introduction to number theory lecture notes adam boocher 20145, edited by andrew ranicki 20156 december 4, 2015 1 introduction 21. Depending on your motivation and degree of comfort reading proofs at this level, something like this might be a good option an introduction to proofs book isnt a necessity for everyone. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. Proof by induction is also presented as an efficient method for proving several theorems throughout the book. On the platonic level, this is intended to communicate something about proof, sets, and logic.
In this book, all numbers are integers, unless specified otherwise. We proceed to define the con cept of divisibility and. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by nonmajors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. This is being written as a textbook for math 502, logic and set theory, and math 522, advanced set theory, at boise state university, on the practical level. 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. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics. Assert the truth of one or more statements and call them axioms or postulates. What are some good introductory books on mathematical proofs. For example, here are some problems in number theory that remain unsolved. Proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. I hadnt realized that i hadnt written it in the notes.
In this book the author solves the problem of maintaining the interest of students at. The problems are culled from various sources, many from actual contests and olympiads, and in general are very di. Number theory and combinatorics indian academy of sciences. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. 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. For example, if you are interested in number theory, you can read harold starks an introduction to number theory. Proof we show the existence of a factorisation into primes by induction.
Number theory, known to gauss as arithmetic, studies the properties of the integers. We consider the socalledmersenne number 2 p 1 and show that any. Wiless proof uses sophisticated mathematical techniques that we. Find all numbers of the form 2nl with positive integer n, not exceeding million, which are products of two primes, and prove that if n. The discussion of the prime number counting function. Number theory and its applications ii, hailong li, fuhuo li, shigeru kanemitsu, world scientific, 2018 linear forms in logarithms and applications, yann bugeaud, irma lectures in mathematics and theoretical physics vol. Number theory school of mathematical sciences queen mary. There is an indication of a considerable simpli cation of my forcing treatment, by restricting the construction of. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second.
The following theorems illustrate a number of important properties of divisibility. By the fundamental theorem of arithmetic, n is divisible by some prime p. An introduction to proofs, logic, sets, and numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures.
1435 1053 1324 402 1190 177 157 255 1528 1365 1407 1000 1513 630 419 900 1496 1173 586 288 1429 1045 927 393 1210 277 170 982 456 137 242 139 1048 580 3 171 26 724 1420 598 496 207