Recall that a prime number is an integer n, greater than 1, such that the only positive integers that evenly divide n are 1 and n. A prime number is a natural number with exactly two distinct divisors. Despite their ubiquity and apparent simplicity, the natural integers are chockfull of beautiful ideas and open problems. Chapter 17 proof by contradiction this chapter covers proofby contradiction. Proof by contradiction often works well in proving statements of the form. Convergence theorems the rst theorem below has more obvious relevance to dirichlet series, but the second version is what we will use to prove the prime number theorem. This is a proof by contradiction, and it is based on the prime factorization of a. Chapter 6 proof by contradiction mcgill university. Proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe.
Basic proof techniques washington university in st. Infinitely many primes one of the first proofs by contradiction is the following gem attributed to euclid. One of the best known examples of proof by contradiction is the provof that 2 is irrational. In proof by contradiction, you assume your statement is not true, and then. Proof by contradiction means that you take a statements negation as a starting point, and find a contradiction. Letting pn denote the number of primes p b n, gauss conjectured in the early. Euclid described in his elements, an ancient greek proof that. Proof by contradiction california state university, fresno. In such a proof, we assume the opposite of what we. I need to prove there are infinite many prime number by contradiction.
Assume that there is an integer that does not have a prime fac torization. Proposition there are infinitely many prime numbers. Demonstrate, using proof, why the above statement is correct. A level maths new specification proof by contradiction answers. This conditional statement being false means there exist numbers a and b for which a,b. Assume that there is an integer that does not have a prime factorization. Therefore, a can be represented as the ratio of two integers. A natural number greater than 1 that is not prime is called a composite number. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. Zagier dedicated to the prime number theorem on the occasion of its 100th birthday the prime number theorem, that the number of primes 1 that is not prime factors in a unique way into a nondecreasing product of primes. A primary focus of number theory is the study of prime numbers, which can be. The proof well give dates back to euclid, and our version of his proof uses one of the oldest tricks in the book and the book. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false.
Its a principle that is reminiscent of the philosophy of a certain fictional detective. This method of proof was so well established in greek mathematics that many of euclids theorems and most of archimedes important results were proved by contradiction. Proof techniques stanford university computer science. We proceed by contradiction, and so we assume that there are finitely many primes. Whenever we have assumptions stating the existence. Proof by contradiction we now introduce a third method of proof, called proof by contra. Notes for lecture 3 1 there are infinitely many primes people. Let p be a finite set whose elements are prime numbers. The volume opens with perhaps the most famous proof in mathematics.
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. Version 1 suppose that c nis a bounded sequence of. But the previous lemma says that every number greater than 1 is divisible by a prime. Recall that a positive integer is said to be prime if it has precisely two positive divisors which are necessarily 1 and the number itself an integer greater than 1. Prime numbers number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. Every positive integer can be written as a product of primes possibly with repetition and any such expression is unique up to a permutation of the prime factors. In a previous video, we used a proof by contradiction to show that the square root of 2 is irrational. This video proves that there are infinitely many prime numbers, using proof by contradiction. For the sake of contradiction, suppose there are only. Negating the two propositions, the statement we want to prove has the form. Pdf we present a new simple proof of the infinitude of primes.
The theorem could be restated as there is no largest prime or there is no finite list of all primes. A simple proof of the prime number theorem 2 a simple proof of the prime number theorem is constructed from mean value theorem for arithmetic functions, and basic properties of the zeta function. One of the oldest mathematical proofs, which goes all the way back to euclid, is a proof by contradiction. So this is a good situation for applying proof by contradiction. If n were prime, it would have an obvious prime factorization n n. This is a contradiction, since we claimed that m 2 2n, and no number can have both an even number of prime factors and an odd number of prime factors. It is a particular kind of the more general form of argument known as reductio ad absurdum. Maybe i wrote the contradiction wrong, but i thought it came out to. If you are trying to factor a number n, you do not need to try dividing by all the numbers from 1 to n. Now any square number x2 must have an even number of prime factors. Helena mcgahagan lemma every integer n 1 has a prime factorization.
We will show that there are infinitely many primes. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. We now introduce a third method of proof, called proof by contra diction. There are no natural number solutions to the equation x2 y2 1. As far as this proof is concerned, for all we know, both composite and prime numbers could be. Mat231 transition to higher math proof by contradiction fall 2014 9 12. In the 17th century, proofs by contradiction became the focus of attention. The difference of any rational number and any irrational number is irrational. A prime number or a prime is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Here is an example of a simple theorem and a simple direct proof.
Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. Assume there are a finite number of prime numbers, that we write as. This proof, and consequently knowledge of the existence of irrational numbers, apparently dates back to the greek philosopher hippasus in the 5th century bc. However, since n2 must also have an even number of prime factors, and 2 is a prime number, 2n2 must have an odd number of prime factors.
Read the proof of the irrationality of the square root of 2 in the introduction for an example. This proof does not require any deep knowledge of the prime numbers, and it does not require any di. The distribution of prime numbers andrew granville. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. Assume we have a set of finitely many primes of the form. Proof by contradiction is often the most natural way to prove the converse of an already proved theorem. In proof by contradiction, we show that a claim p is true by showing that its negation p. Hardy described proof by contradiction as one of a mathematicians finest weapons, saying it is a far finer gambit than any. The idea of proof by contradiction is quite ancient, and goes back at least as far as. Proof that there are infinitely many prime numbers euclid. A prime number is a number that has exactly two divisors.
This is euclids proof that there are infinitely many prime numbers, and does indeed work by contradiction. Since 2 is a prime number, the list of p is is nonempty. 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. I also dont really understand how to do these cases using contradiction. The zeta function is an important function in mathematics. This contradicts the assumption that q is the smallest positive rational number. What i want to do in this video is essentially use the same argument but do it in a more general way to show that the square root of any prime number is irrational.
Shorser the following proof is attributed to eulclid c. If p is a prime number bigger than 2, then p is odd. If p is nite, then by the proposition, p is not the set of all primes. Simple proof of the prime number theorem january 20, 2015 2. On the analysis of indirect proofs example 1 let x be an integer. If a friend says the primes are not infinite theres the contradiction, then you could say. A prime number is a positive integer which has no positive integer factors other than 1 and itself. We take the negation of the theorem and suppose it to be true. This is the simplest and easiest method of proof available to us.
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