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V. Adamchik 1 Integer Divisibility Victor Adamchik Fall of 2005 Lecture 1 (out of seven) Plan 1. Basics of divisibility 2. Prime numbers 3. Perfect numbers Notations - set of integers - set of positive integers (also , also ) - set of nonnegative integers (also 0 ) - logical AND - logical OR - exist (existential quantifier) ! - exist exactly one (unique existential quantifier) - any (universal quantifier) x - integer part of x (or the floor function) Basics of divisibility In this chapter, we will discuss the divisibility of integers, the set of integers is denoted by . We will give a few detailed proofs of some of the basic facts about divisibility. Most of the properties are quite obvious, but it is still a good idea to know how to prove them. Definition. An integer b 0 divides another integer a iff k that a k b. V. Adamchik 21-127: Concepts of Mathematics We also say that b is a factor (or divisor) of a. One frequently writes b Example. 3 12 Exercise. a to indicate that b divides a. but 5 12 Let a and b be positive integers and a b. How many positive integers not exceeding a are divisible by b? In other words, find such c that b Solution. All numbers divisible by b are in the form b k, where k c a and b c . They are positive and do not exceed a, 0 Therefore, there are floor( ab ) or b a b k a or 0 k a b such integers Theorem 1. For all integers a, b, c (1) 1 a, 1 a and a 0. (2) Reflexivity: a a. (3) Transitivity: a b b c a c. (4) Not-quite antisymmetry: a b (5) if a b a c b a a n b a b a b. m c for any integers n and m Proof. (1) and (2) follow immediately a (3) Given b Then c y x a and c b (4) Given b Then a y xy a x a a 1 a 1 a a y b. x x a and a and therefore x y or y x a y a, so a | c y b. xy 0, 1 (there are no zero-divisors in the integers). It follows that either y 1. But x 1 implies a b, and x 1 implies b a. x 1 V. Adamchik 3 (5) Given b x a and c Consider n b m y a. c nb mc xan y am a xn ym a nb mc It follows a nb mc Application of Theorem 2. Do there exist integers x, y, and z such that 6 x 9y 15 z 107? No, they don't, here is the proof by contradiction. Since 3, 6 and 9 has a common divisor 3 than 3 must divide its linear combination 3 6x 9y 15 z 3 107 which is wrong. Question. How many divisors does a positive integer have? Here is a picture of all divisors of integers in range [1, 500] 20 15 10 5 100 200 300 400 500 Primes Observation. Every positive integer has at least two divisors: 1 and itself Definition. Integer p 1 is called a prime if its only positive divisors are 1 and p. Otherwise it is called a composite. V. Adamchik 21-127: Concepts of Mathematics The number 1 is a special case which is considered neither prime nor composite The number 2 is also special, it is the only even prime. Theorem. There are an infinite number of primes Proof. (by contradiction) Assume otherwise, say, p1 , …, pn is a complete list of all primes. Define p p1 p2 … pn 1. Since this number p is larger than all the pi , it cannot be prime. But then, there is some prime that divides p. Since our list is supposedly complete that prime must be, say, pr . We have that pr pr p p pr pr p p1 p2 … pn pr 1 p1 p2 … pn But then pr 1. A contradiction. QED - end of proof ("quod erat demonstrandum"). How would you find (or generate) primes? Sieve of Eratosthenes: (Greek astronomer, 195BC) Write down the integers from 2 to the highest number n you wish to include in the table. Cross out all numbers 2 which are divisible by 2. Cross out all numbers 3 which are divisible by 3, then by 5 and so on. Continue until you have crossed out all numbers divisible by n . V. Adamchik 5 Why do we stop at n ? Because the next number to cross must be n since we cross all numbers with divisors n . Goldbach Conjecture (1742) Prime numbers satisfy many strange and wonderful properties. Observation: 6 7 8 9 10 18 3 3 5 2 5 3 7 2 7 3 11 7 What about 117 ? Can you represent it as sum of two primes? Goldbach made the conjecture that every odd number > 5 is equal to the sum of three primes. Euler replied that Goldbach's conjecture was equivalent to the statement that every even number > 2 is equal to the sum of two primes. p1 p2 2n It is known to be true for for numbers through 6 1016 (checked numerically in 2003) Mersenne numbers For some years, people believed that if p is prime, then so is 2 p 22 1, 23 1, 25 1: 1, ... This is not true for all primes, for example 211 1 2047 23 89 Mersenne Conjecture (15 century, by French monk Marin Mersenne) 2 p p 1 is prime for 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 , 257 and composite for all other positive integers p < 257. V. Adamchik 21-127: Concepts of Mathematics It took a few of centuries to show that the conjecture was wrong. Only in 1947 the range up to 258 was checked! It turned out that a) 267 1 and 2257 1 are not primes b) Mersenne missed p Definition: When 2 p 61, 89, 107. 1 is prime it is said to be a Mersenne prime. The largest known Mersenne prime is (2005): 225,964,951 Theorem. If 2 p 1 is prime, then p is prime. Proof. By contradiction - we assume that 2 p number, p 1 1 is prime, but p is not prime. Let p be a composite r s. Consider the following polynomial xr s 1 It can be written as xr s 1 xs 1 xs r 1 xs r 2 ... xs 1 which is easily proved by expanding the right hand side. Therefore, if p is composite then x p Contradiction to our assumption that 2 p 1 is composite, so is 2 p 1, since it's divisible by 2s 1. 1 is prime. QED. Perfect numbers Definition. An positive integer is a perfect number if it equals the sum of its proper divisors (not including itself). The first few perfect numbers are 6=1+2+3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 8128 = ... V. Adamchik 7 Question. What is the next perfect number? It seems it should not be a problem to answer this by writing Java or C program. Question. Are they all even? This question is much much harder.... It is not known if any odd perfect numbers exist. All even perfect number are in the form 6=1+2+3 =2*3 28 = 1 + 2 + 3 + 4 + 5 + 6 +7 = 4 * 7 496 = 1 + 2 + 3 +... + 31 = 16 * 31 8128 = 1 + 2 + 3 +... + 127 = 64 * 127 Generally, 2n 1 2n 1 , when n is prime This is a relation between the perfect and the Mersenne primes. So the search for Mersenne primes is also the search for even perfect numbers!