Number theory proof by contradiction pdf

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. Many of the statements we prove have the form p q which, when negated, has the form p. Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some e ort needs to be put into veri cation of such a proposed proof. Consequently 1 is not an upper bound for n if it were, since 1 number theorem, a brief history. This is really a special case of proof by contrapositive where your \if is all of mathematics, and your \then is the statement you are trying to prove. More formally, for a 6 0 we say that divides b if there is.

An integer n 1 is called a prime number if its positive divisors are 1 and n. A prime number is a positive integer which has no positive integer factors other than 1 and itself. This is really a special case of proof by contrapositive where your \if is all of mathematics, and your. Assume p, and then use the rules of inference, axioms, defi nitions, and logical equivalences to prove q. Proofs in number theory, algebra, geometry and analysis.

The converse of the pythagorean theorem the pythagorean theorem tells us that in a right triangle, there is a simple relation between the two leg lengths a and b and the hypotenuse length, c, of a right triangle. Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. Mat231 transition to higher math proof by contradiction fall 2014 5 12. 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. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. To prove that p is true, assume that p is not true. Consequently 1 is not an upper bound for n if it were, since 1 number theory 719 congruence modulo i in number theory, we often care if two integers a. Proof techniques stanford university computer science. It covers the basic background material that an imo student should be familiar with. Discrete mathematics introduction to number theory 119 introduction to number theory i number theory is the branch of mathematics that deals with integers and their properties i number theory has a number.

So again, there is a decimal number that is equal to an number the conclusion is false even number your turn using a proof by contradiction approach, investigate the following statement. Just as the proof above for 2, this shows that p divides n which means that p2 divides n2. Divisibility is an extremely fundamental concept in number theory, and has applications including. Besides number theory proof by contradiction may be applied in many areas including geometry, linear algebra and calculus. Use the method of proof by contradiction to prove the following statements. I hadnt realized that i hadnt written it in the notes. In each case you should also think about how a direct or contrapositive proof. The statement \a implies b can be written symbolically. Proof by contradiction is common in number theory because many proofs require some kind of binary choice between possibilities. There are a number of major changes in the fourth edition. Apr 03, 2018 former mcdonalds worker does a number theory proof duration. We arrive at a contradiction when we are able to demonstrate that a statement is both simultaneously true and false, showing that our assumptions are inconsistent. This shows that p divides m2, which again shows that p divides m. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures.

Proof by contradiction this is an example of proof by contradiction. Proof by contradiction also known as indirect proof or the method of reductio ad absurdum is a common proof technique that is based on a very simple principle. For an explanation of the background pattern, skip ahead to the end of the page. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Im not really sure what im trying to do from there. Methods of proof direct, contradiction, conditional, contraposition. This chapter will introduce many important concepts that will be used throughout the rest of the book. The orange ball marks our current location in the course. For example, here are some problems in number theory that remain unsolved. Dear jamie, supreme integral with feynmans trick duration.

Contents preface ix introduction x i fundamentals 1. This tool is the principle of mathematical induction. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematical proofs. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. We start with the language of propositional logic, where the rules for proofs are very straightforward. Lets start by asserting that the opposite is true, viz. R and r has the least upper bound property, then n has a least upper bound. It would be tempting to claim that the above theorem claims that the. Assume, by way of contradiction, that there are a nite number of primes p 1. Introduction to analytic number theory fall 2015 if you find a mistake, omission, etc. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. If it were rational, it would be expressible as a fraction ab in lowest terms, where a and b are integers, at least one of which is odd. Sometimes the negation of a statement is easier to disprove leads to a contradiction than the original statement is to prove. Proof theory is not an esoteric technical subject that was invented to support a formalist doctrine in the philosophy of mathematics.

Proofs and concepts the fundamentals of abstract mathematics by dave witte morris and joy morris university of lethbridge incorporating material by p. 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 readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. 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. Proof by contradiction a proof by contradiction is a proof that works as follows.

This prime p must be among the p i, since by assumption these are all the primes, but n is seen not to be divisible by any of the p i, contradiction. Common mistakes in proofs show that if x is real number, then x2 is positive. This is euclids proof that there are infinitely many prime numbers, and does indeed work by contradiction. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. These perfect numbers have seen a great deal of mathematical studyindeed, many of the basic theorems of number theory stem from the investigation of the. There is an indication of a considerable simpli cation of my forcing treatment, by restricting the construction of. I more technically, if a and b are integers and m a positive integer, a b mod m i m ja b. Both fermats last theorem and the modularity theorem were almost universally considered inaccessible to proof by. There is a new chapter on mathematical induction chapter 26. Eulers theorem is a generalization of fermats little theorem and the proof presented here has the same form as the proof presented above for fermats little theorem.

In this exercise, we prove that the square root of 3 is an irrational number. 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. All we were told is 1 to do a proof by contradiction, and 2 to use the cases i. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. Relation between proof by contradiction and proof by contraposition to rewrite the proof as a proof by contradiction, you suppose there is an x in d such that px and qx.

Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy. Wiless proof of fermats last theorem is a proof by british mathematician andrew wiles of a special case of the modularity theorem for elliptic curves. This leads to a contradiction with the fact that there are s objects for sk. This means that 2 can be written as the ratio of two integers, a and b such that 2 equation 1. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. By the fundamental theorem of arithmetic, n is divisible by some prime p. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. The point of departure for a proof by contradiction is the assumption that the statement to be proved is false.

Division given two integers, say a and b, the quotient ba may or may not be an integer e. Olympiad number theory through challenging problems. In particular, you can certainly find lots of books written by perfectly good mathematicians in which the proof is described as a proof by contradiction. So this is a valuable technique which you should use sparingly. Then, the book moves on to standard proof techniques. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. Suppose for the sake of contradiction that it is not true that 2 is irrational. Still, there seems to be no way to avoid proof by contradiction. 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.

To prove a statement p is true, we begin by assuming p false and show that this leads to a contradiction. Chapter 6 proof by contradiction mcgill university. It is used in the proof that a polynomial of degree d has at most d roots modulo p. We take the negation of the theorem and suppose it to be true. So, there is an odd number that is equal to an even 5. Assume for the sake of contradiction that s fc2z j0 proofs, i found it somehow unsatisfying that a proof by contradiction doesnt end by sort of unrolling the contradiction backward, along the lines of this is a contradiction, so the statement before it must be false, which means the statement before that must be false and so on, until wed worked our way back to the original assumption. The proof by contradiction method makes use of the equivalence p p f 0 where f 0 is any contradiction. Using the same letter to mean two different things. In the above proof we got the contradiction bis even is even which has the form c in general, your contradiction need not necessarily be of this form. Based on the assumption that p is not true, conclude something impossible. A direct proof, or even a proof of the contrapositive, may seem more satisfying.

The book starts with the basics of set theory, logic and truth tables, and counting. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Suppose that none of the boxes contains more than one object. Hardy described proof by contradiction as one of a mathematicians finest weapons, saying it is a far finer gambit than any. Introduction to analytic number theory tu chemnitz.

An introduction to proof by contradiction, a powerful method of mathematical proof. Another useful dose of maths for everyone by dr sarada. Ninghui lis slides divisibility definition given integers a and b, b. It will actually take two lectures to get all the way through this. In proof by contradiction, we show that a claim p is true by showing that its negation p. The sum of two even numbers is not always even that would mean that there are two even numbers out there in the world somewhere thatll give us an odd number when we add them. Anderson and greg welty what is the relationship between the laws of logic and the existence of god. Practice number theory problems 3 b show that if a b mod n, then for all positive integers c, ac bc mod n. We use a proof by contradiction and assume that the square root of 3 is actually a rational number, and we set it up. Relation between proof by contradiction and proof by contraposition as an example, here is a proof by contradiction of proposition 4. Preface inwriting this book i have been motivated by the desire to create a highqualitytextbookthatcostsalmostnothing.

This fact is then used in place of primitive roots as a tool to. For every positive rational number x, there is a positive rational number y for which y proof. Lecture notes 3 iv before we start our proof, we want to point out that this statement is a generalization of the previous one. Some material on proof by contradiction has been moved forward to chapter 8.

The proof of independence of ch by forcing is added. Proof reasoning by contradiction, assume n is bounded from above. If we were formally proving by contradiction that sally had paid her ticket, we would assume that she did not pay her ticket and deduce that therefore she should have got a nasty letter from the council. Let set sbe the set of numbers which cannot be written as a product of primes. Math 248 methods of proof in mathematics 4 units prerequisite. In this chapter, we will explore divisibility, the building block of number theory. In this case, we implicitly assume all of number and set theory to tackle the problem. This is a contradiction, because m and n have no common divisors. These notes serve as course notes for an undergraduate course in number the ory. The proof by contradiction is grounded in the fact that any proposition must. The difference of any rational number and any irrational number is irrational. This conditional statement being false means there exist numbers a and b for which. Definition any integer number n 1 that is not prime, is called a composite number.

From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. Overview of number theory basics university of texas at. Four basic proof techniques used in mathematics youtube. If we compute mod n on both sizes, nqc cancels out and we obtain ac bc mod n. You then follow the steps of the proof by contraposition to deduce the statement px.