Since the product of odd numbers is odd, ab is odd. Just as the proof above for 2, this shows that p divides n which means that p2 divides n2. Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs reinforces the foundations of calculus and algebra explores how to use both a direct and indirect proof to prove a theorem presents the basic properties of real numbers. Oct 19, 2016 the interested reader can refer to the book proofs from the bookthere is an interesting reason behind this intriguing title, which has the six famous proofs of the infinitude of primes. There is an indication of a considerable simpli cation of my forcing treatment, by restricting the construction of. This 19751987 book by gaisi takeuti 19262017, who apparently died just 3 weeks ago 2017510 according to wikipedia, is a heavyweight book on proof theory at the graduate level or higher. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true.
We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. Since q2 is an integer and p2 2q2, we have that p2 is even. B the book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. A proof by construction that uses cases and an indirect. To rewrite the proof as a proof by contradiction, you suppose there is an x in d such that p x and q x. Contributions to automorphic forms, geometry, and number theory, 521560, johns hopkins univ. Exactly the same sequence of steps can be used as the heart of a proof by contradiction for the given statement. Number theory is replete with sophisticated and famous open problems. This implies that there are integers u and v such that m u2 and n v2. In a proof by contradiction, we start by assuming the opposite, p. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. God made the integers, all else is the work of man. Then there exist two integers, n and m with no common divisor such that v p nm. 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.
Contradiction and contraposition in a direct proof you start with the hypothesis of a statement and make one deduction after another until you reach the conclusion. One of the most common hash functions uses modular arithmetic. For p q to be true both p and q must be true, otherwise it. The systematic study of number theory was initiated around 300b. An introduction to higher mathematics whitman people. Free discrete mathematics books download ebooks online. One kind of indirect proof, argument by contradiction, is based on the fact that. For any natural number n, the sum of all natural numbers less than n is not equal to n.
Relation between proof by contradiction and proof by contraposition to do this, you suppose you are given an arbitrary element x of d such that qx. The title of the book, topology of numbers, is intended to express this visual slant, where we are using the term topology with its. Consumers maximize utility ux,y which is increasing in both arguments and quasiconcave in x,y. This is being written as a textbook for math 502, logic and set theory, and. In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. A course on number theory pdf 9p this note explains the following topics. By contrast, euclid presented number theory without the flourishes. Greek number theory was exclusively interested in natural numbers. I do use chapter 3 combinatorics and add on some graph theory later on in the course. The second and the third proof use special wellknown number. An introduction to higher mathematics download link. One such method is known as an indirect proof or a proof by contraction. Pdf doc del clipboard journal article mr20586182005g. Direct proof example 2 show that if m and n are both square numbers, then m n is also a square number.
Given a natural number n, if n is prime, then it is the product of just one prime. Let us assume that there are two different prime factorizations of n, say n p e 1 1 2 2 e m m n q d 1 1 2 2 d r r where q 1 book is designed for a one semester course in discrete mathematics for sophomore or junior level students. This book offers an introduction to the art and craft of proof writing. Learn exactly what happened in this chapter, scene, or section of geometric proofs and what it means. An alternative strategy for proving a b called indirect proof or. But that contradicts the assumption that r was the smallest rational number if r is the smallest rational number were q, then one can infer from r 2 is a. This book is a concise introduction to number theory and some related algebra, with an emphasis on solving equations in integers. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. It covers the basic background material that an imo student should be familiar with. The conjunction of the formulas p and q is the formula \p and q, which is written symbolically as p q. In case 2, we use an indirect proof to show that p n. Still, there seems to be no way to avoid proof by contradiction. The idea is that the conclusion c is a consequence of the hypotheses h 1, h.
Euclidean and noneuclidean geometry have a common basic set. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Division given two integers, say a and b, the quotient ba may or may not be an integer e. When trying to construct a proof it is sometimes useful to assume the opposite of the thing you are trying to prove, with a view to obtaining a contradiction. The prime number theorem for arithmetic progressions ii 2 38 16.
This book provides an introduction to number theory from a point of view that is more geometric than is usual for the subject, inspired by the idea that pictures are often a great aid to understanding. An introduction to higher mathematics by patrick keef, david guichard, russ gordon. The following simple but wonderful proof is at least as old as euclids book the elements. He proved the fundamental theorems of abelian class. Theorems, corollaries, lemmas, and methods of proof wiley. Z to be an open set if and only if it is either the empty set. Let us assume that there are two different prime factorizations of n, say n p e 1 1 2 2 e m m n q d 1 1 2 2 d r r where q 1 proof, or even a proof of the contrapositive, may seem more satisfying. Some of his famous problems were on number theory, and have also been in. But compared with infinity, any number, no matter how large, is less than a drop in the bucket. Indirect proof, also called proof by contradiction, assumes the hypothesis if given together with a negation. The purpose of this book is to present a collection of interesting problems in elementary number theory. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms. Number theory has a long and distinguished history and the concepts and problems relating to the subject have been instrumental in the foundation of much of mathematics.
The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. It covers the basic background material that an imo student should be familiar. As in earlier books, euclid probably based much of his work on the discoveries of. Boolean functions and computer arithmetic, logic, number theory and cryptography, sets and functions, equivalence and order, induction, sequences and series. A number that is not rational is called irrational and cannot be written as a fraction, p q, where p and q are both integers. The prime number theorem and the riemann hypothesis 3 1. Contents preface vii section1integers 1 2uniquefactorization 10 3lineardiophantineequations 20 4congruences 27 5linearcongruences 34 6fermatsandwilsonstheorems 42 7thedivisorsofaninteger 49 8perfectnumbers 56 9eulerstheoremandfunction 63 10primitiverootsandindices 72 11quadraticcongruences 82 12quadraticreciprocity 92. In earlier versions of our logic style manual, we tended to. Finding integer solutions led to two fundamental ideas of number theory in ancient times the euclidean algorithm and unique prime factorization and in modern times to two fundamental ideas of algebra rings and ideals. Number theory the euclidean algorithm, the fundamental theorem of arithmetic. Proof, sets, and logic department of mathematics boise state. Larsens alternative, moments, and the monodromy of lefschetz pencils. A primary focus of number theory is the study of prime numbers, which can be. Contents preface vii introduction viii i fundamentals 1.
Number theory is a beautiful branch of mathematics. The natural numbers grow beyond all bounds, and every natural number n 2 has a prime divisor. Each section contains a representative selection of problems. Prime numbers number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. We will see, that even when the original problem involves only ordinary. Number theory is of course primarily the theory of numbers. A good one sentence answer is that number theory is the study of the integers, i. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. In the 1950s, hillel furstenberg introduced a proof by contradiction using pointset topology. 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. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Let us assume that there are two different prime factorizations of n, say n p e 1 1 2 2 e m m n q d 1 1 2 2 d r r where q 1 proof by contradiction and proof by contraposition proof by contraposition.
He later defined a prime as a number measured by a unit alone i. Between its publication and andrew wiless eventual solution over 350 years later, many mathematicians. Proof of the fundamental theorem on finite commutative groups 258 6. Despite their ubiquity and apparent simplicity, the natural integers are chockfull of beautiful ideas and open problems. What is the best textbook for selflearning number theory. Algebraic numbers, finite continued fractions, infinite continued fractions, periodic continued fractions, lagrange and pell, eulers totient function, quadratic residues and nonresidues, sums of squares and quadratic forms.
Oct 28, 2014 indirect proof is a type of proof in which a statement to be prov ed is assumed false and if the assumption leads to an imp ossibility, then the statement assumed false has b een proved to be true. Qa241sil these books are both excellent sources of examples, additional practice problems and i. 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. But k n is an integer because it is a difference of integers.
Number theory a lively introduction with proofs, applications, and stories pdf. Consumer theory and the envelope theorem 1 utility maximization problem the consumer problem looked at here involves two goods. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate. These are usually polynomial equations with integral coe. Olympiad number theory through challenging problems. Qa241ros a friendly introduction to number theory by j. Elementary number theory dudley provides a very readable introduction including practice problems with answers in the back of the book. Now, r 2 is a rational number greater than 0 and smaller than r. Relation between proof by contradiction and proof by contraposition proof by contraposition. A simple, constructive proof, based mainly on euclid, book iii, is given in. Number theory euclids number theory the overall structure elements viiix as in earlier books, euclid probably based much of his work on the discoveries of others, but the organization and presentation was his own. An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs. He began book vii of his elements by defining a number as a multitude composed of units.
Chapter 5 indirect proofs there are times when trying to prove a theorem directly is either very difficult or impossible. As i add sections, the table of contents will get longer and eventually change the page numbering of the later content in terms of the pdf. Indirect proof is a type of proof in which a statement to be prov ed is assumed false and if the assumption leads to an imp ossibility, then the statement assumed false has b een proved to be true. What are the \objects of number theory analogous to the above description. Attempts to do so have led to the strange world of constructive mathematics. The text covers the mathematical concepts that students will encounter in many disciplines such as computer. Hence m n equals 2 times an integer, and so by definition of even, m n is even.
The proof of independence of ch by forcing is added. As in earlier books, euclid probably based much of his work on the discoveries of others, but the organization and presentation was his own. Direct proofs of lehmussteiners theorem are proposed. The symbol oindicates part of a proof is done, but it continues. When that occurs, we rely on our logic, our everyday experiences, to solve a problem. Besides reading the book, students are strongly encouraged to do all the. Proofs direct proofs, existence proofs, mathematical induction, indirect proof. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. This textbook takes a problemsolving approach to number theory, situating each theoretical concept within the framework of some examples or some problems for readers.
Proof we show the existence of a factorisation into primes by induction. An important aspect of number theory is the study of socalled diophantine equations. Thus, the prerequisites for this book are more than the prerequisites for most elementary number theory books, while still being aimed at undergraduates. Thus, i would say it does a very nice job of both introducing students to proof and to intro number theory and combinatorics. When reading mathematical books you will notice that the word implies appears more often than. The fact that a proof in number theory exists that uses topology, says something about the foundational role of number theory. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Directions for writing proofs of universal statements writing proofs is similar to writing a computer program based on a set of specifications. We also find number theory intrinsically interesting, one of the most beautiful subjects in modern mathematics, and all the more interesting because of its roots in antiquity. Apart permutation groups and number theory, a third occurence of group theory which is worth mentioning arose from geometry, and the work of klein we now use the term klein group for one of the groups of order 4, and lie, who studied transformation groups, that is transformations of geometric objects. This technique is known as indirect proof or proof by contradiction. In this book, professor baker describes the rudiments of number theory in a concise, simple and direct manner. When i refer to page number, it will be the document numbering, not the pdf numbering. This book covers all of the needed proof techniques and gives interesting examples for them.
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