Prime and Perfect Numbers

Transcription

1 Prime and Perfect Numbers 0.3 Infinitude of prime numbers Euclid s proof Euclid IX.20 demonstrates the infinitude of prime numbers. 1 The prime numbers or primes are the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,... (1) which cannot be resolved into smaller factors We have to prove that there are infinitely many primes, i.e., that the series (1) never comes to an end. Let us suppose that it does, and that 2, 3, 5,..., P is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number Q defined by the formula Q = P + 1. It is plain that Q is not divisible by any of 2, 3, 5,..., P ; for it leaves the remainder 1 when divided by any one of these numbers. But, if not itself prime, it is divisible by some prime, and therefore there is a prime (which may be Q itself) greater than any of them. This contradicts our hypothesis, that there is no prime greater than P ; and therefore this hypothesis is false Euler s proof Suppose, for a contradiction, that there be only finitely many many primes p 1, p 2,...,p N. By the Fundamental Theorem of Arithmetic, n + 1 The presentation is taken from G. H. Hardy, A Mathematician s Apology, pp There are technical reasons for not counting 1 as a prime.

2 8 CONTENTS = = ( ) ( p 1 p ) ( 1 p 2 p p p p N, p N + 1 contradicting the divergence of the harmonic series n Some important theorems on prime numbers p 2 N ) + Theorem 0.1 (Tchebycheff). 3 For every integer k 2, there exists a prime number p in the range k < p < 2k. Theorem 0.2 (Dirichlet ( )). If a and b are relatively prime, the arithmetical progression an + b contains infinitely many prime numbers. Theorem 0.3 (The Prime Number Theorem). 4 If π(x) denotes the number of primes between 1 and x, then π(x) as x. x log x Some unsolved problems about prime numbers 5 1. (Goldbach conjecture) 6 Every even number > 6 is a sum of two distinct odd primes. 2. (Twin prime conjecture) There are infinitely many pairs of prime numbers of the form p and p Does the sequence n contain infinitely many primes? 0.5 Perfect numbers A number is perfect if it is equal to the sum of all its divisors, including 1 but excluding the number itself. The first five even perfect numbers are 6, 28, 496, 8128, For each positive integer n, let σ(n) denote the sum of all positive divisors of n, including 1 and n itself. A number n is a perfect number if and only if σ(n) = 2n. 3 This is also known as the Bertrand postulate. It was conjectured by J.L.F. Bertrand ( ) in 1845, and was proved by P.L. Tchebycheff ( ) in 1850). 4 The prime number theorem was conjectured by A.M. Legendre ( ) and C.F Gauss ( ) around 1800, first proved at the end of the 19th century by J. Hadamard ( ) and de la Vallée Poussin ( ) using deep analytical methods, but by elementary (but difficult) methods by Paul Erdös ( ) and A. Selberg in For more unsolved problems about prime numbers, see Chapter 1 of R.K. Guy. Unsolved Problems in Number Theory, (2nd ed.), Springer Verlag, Proposed to Euler in 1742 by C. Goldbach ( ). 7 As of 1990, the largest known pair of twin primes are ± 1.

3 0.6 Mersenne primes 9 Theorem 0.4 (Euclid IX.36). If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. Modern formulation: Suppose M k := k 1 = 2 k 1 is prime. Then 2 k 1 (2 k 1) is a perfect number. (Note that such a perfect number is even). Proof. σ(2 k 1 (2 k 1)) = σ(2 k 1 )σ(2 k 1) = (2 k 1) 2 k = 2 2 k 1 (2 k 1). Theorem 0.5 (Euler). Every even perfect number is of the form described in Eucl.IX.36: If n is an even perfect number, then n = 2 k 1 (2 k 1) for some integer k and M k = 2 k 1 is prime. Proof. Write n = 2 k 1 q, q odd. Since n is perfect, 2 k q = 2n = σ(n) = σ(2 k 1 )σ(q) = (2 k 1)σ(q). From this, σ(q) = q + q 2 k 1. Since σ(q) is an integer, 2k 1 must be a divisor of q. Indeed, 2 k 1 = q, and σ(q) = q + 1. This means that q = 2 k 1 is a prime. Remark. It is not known if an odd perfect number exists. 0.6 Mersenne primes The number M k := 2 k 1 is called the kth Mersenne number. If M k is prime, then k must be a prime. (Exercise). For example, M 3 = 7, M 5 = 31 and M 7 = 127 are primes, but M 11 = = 2047 is not a prime. (Exercise. Factorize it) Primality tests for M k Theorem 0.6 (Fermat). If k is a prime, then every prime factor of M k = 2 k 1 must be of the form 2kr + 1 for some integer r. Remark. In the early 20-th century, F.N.Cole, Professor of Mathematics in Columbia University, spent the sundays of three consecutive years on the factorization of M 67, and obtained M 67 = Theorem 0.7 (Lucas - Lehmer). Let (v k ) be the sequence defined recursively by v i+1 = v 2 i 2, v 2 = 4. For a prime number k, the Mersenne number M k = 2 k 1 is prime if and only if M k divides v k. Remark. It is not known if there are infinitely many Mersenne primes, (equivalently even perfect numbers).

4 10 CONTENTS Records of Mersenne primes k Year Discoverer k Year Discoverer 2 Ancient 3 Ancient 5 Ancient 7 Ancient 13 Ancient P.A.Cataldi P.A.Cataldi L.Euler I.M.Pervushin R.E.Powers E.Fauquembergue E.Lucas R.M.Robinson R.M.Robinson R.M.Robinson R.M.Robinson R.M.Robinson H.Riesel A.Hurwitz A.Hurwitz D.B.Gillies D.B.Gillies D.B.Gillies B.Tuckerman C.Noll, L.Nickel C.Noll H.Nelson, D.Slowinski D.Slowinski W.N.Colquitt, L.Welsch D.Slowinski D.Slowinski D.Slowinski,P.Gage D.Slowinski Slowinski and Gage Armengaud, Woltman et al Spence, Woltman, et.al Clarkson et. al Hajratwala et. al Cameron, Woltman, Michael Shafer Findlay Nowak Cooper, Boone et al Cooper, Boone et al /8/ /8/2008 The most recently discovered Mersenne primes M and M have about 11.1 million and 12.9 million digits and are the largest known primes. 0.7 Fermat numbers The number F k := 2 2k + 1 is called the kth Fermat number. Fermat ( ) observed that F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257, F 4 = are all primes, and conjectured that all F k are primes. This was later refuted by Euler ( ) who found the factorization No other Fermat primes are known. F 5 = = = Remarks. (1) Every factor of F k is of the form 2 k+2 a + 1 for some integer a. (2) F. G. Eisenstein ( ) conjectured that , , ,..., are primes. However, it was found out that = F 16 is composite (J. L. Selfridge). Relation with geometry. Gauss ( ) discovered that it is possible to construct (using only ruler and compass) a regular polygon of n sides if n is the product of a power of 2 and distinct Fermat primes. He stated that the converse is also true. This was later proved by P. L. Wantzel ( ). For Gauss construction of the regular 17 gon.

5 0.7 Fermat numbers 11 Exercise (a) q is a prime number if and only if σ(q) = 1 + q. (b) σ(2 k 1 ) = 2 k 1. Determine if M 13 = 8191 and M 23 = are prime numbers. Use the Lucas - Lehmer test for (re-)confirm that M 5 = 31 and M 7 = 127 are primes, but that M 11 = 2047 is not.

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