Introduction Integers Discrete Mathematics Andrei Bulatov
Discrete Mathematics - Integers 9- Integers God made the integers; all else is the work of man Leopold Kroenecker
Discrete Mathematics - Integers 9-3 Division Most of useful properties of integers are related to division If a and b are integers with a, we say that a divides b if there is an integer c such that b = ac. When a divides b we say that a is a divisor (factor) of b, and that b is a multiple of a. The notation a b denotes that a divides b. We write a b when a does not divide b Example. Let n and d be positive integers. How many positive integers not exceeding n are divisible by d? The numbers in question have the form dk, where k is a positive integer and < dk n. Therefore, < k n/d. Thus the answer is n/d
Discrete Mathematics - Integers 9-4 Properties of Divisibility Let a, b, and c be integers. Then (i) if a b and a c, then a (b + c); (ii) if a b, then a bc for all integers c; (iii) if a b and b c, then a c. Proof. (i) Suppose a b and a c. This means that there are k and m such that b = ak and c = am. Then b + c = ak + am = a(k + m), and a divides b + c.
Discrete Mathematics - Integers 9-5 Properties of Divisibility (cntd) If a, b, and c are integers such that a b and a c, then a mb + nc whenever m and n are integers. Proof. By part (ii) it follows that a mb and a nc. By part (i) it follows that a mb + nc. If a b and b a, then a = ±b. Proof. Suppose that a b and b a. Then b = ak and a = bm for some integers k and m. Therefore a = bm = akm, which is possible only if k,m = ±.
Discrete Mathematics - Integers 9-6 The Division Algorithm Theorem (The division algorithm) Let a be an integer and d a positive integer. Then there are unique integers q and r, with r < d, such that a = dq + r d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder Examples: - Let a = and d = Then = 9 + - Let a = - and d = 3 Then - = 3 (-4) + - Let a = 3 and d = Then 3 = + 3
Discrete Mathematics Primes 3-7 Representation of Integers In most cases we use decimal representation of integers. For example, 657 means 6 + 5 + 7 = Let b be a positive integer greater than. Then if n is a positive integer, it can be expressed uniquely in the form k k k k n = a b + a b + L+ a b + a where k is a nonnegative number, a k, ak, K,a, a are nonnegative integers less than b, and a k Such a representation of n is called the base b expansion of n, denoted by (a a Ka a ) k k b 6 + 5 + 7
Discrete Mathematics Primes 3-8 Binary Expansion a Important case of a base is. The base expansion is called the binary expansion of a number k k n = a + a + L+ a + a Find the binary expansion of 65 8 65 a k 4 8 65 = 8 + 8 = 4 + 4 = + k 4 a a a 3 4 a5 a6 = + = 5 + 5 = + 65 = () 5 5 = + = + a 7
Discrete Mathematics Primes 3-9 Hexadecimal Expansion Using A, B, C, D, E, F for,,, 3, 4, 5, respectively, find the base 6 expansion of 7567
Discrete Mathematics Primes 3- Primes Every integer n (except for and -) has at least positive divisors, and n (or -n). A positive number that does not have any other positive divisor is called prime Prime numbers:, 3, 5, 7,, 3, 7, 9, 3, 9, 3, 37, 4, 43, Mersenne numbers are the numbers of the form M = n n There are many prime numbers among Mersenne numbers. The greatest known prime number is M 358657 358657 = 358439369395 The next candidate is = M M6 A positive number that is not prime is called composite
Discrete Mathematics Primes 3- Composite Numbers Every composite number has a prime divisor. Proof. Let S be the set of all composite numbers that do not have a prime divisor Since S N, by the Well-Ordering Principle, it has a least element r. As r is not prime, it has a divisor, therefore, r = uv for some positive integers u and v. u < r and v < r. Therefore u S, and u has a prime divisor p. Since p u and u r, we conclude that p r, a contradiction.
Discrete Mathematics Primes 3- How many prime numbers are there? Theorem (Euclid) There are infinitely many prime numbers. Proof. By contradiction. Suppose that { p,p, K, pk} is the set of all prime numbers, and let a = pp Kpk + Since a is greater than any member of the list, a is composite. By the previous statement, a has a prime divisor, that is for some we have p j a Since p j a and p p p Kp we have p a p p Kp = j k A contradiction. n If n is a positive integer, then there are approximately prime ln n numbers not exceeding n j k p j
Discrete Mathematics Primes 3-3 Open Problems about Primes Goldbach s Conjecture Every positive even number can be represented as the sum of two prime numbers. For example: 4 = +, 8 = 5 + 3, 4 = 37 + 5 Goldbach s conjecture is known to be true for even numbers 7 up to The Twin Prime Conjecture Twin primes are primes that differ by, such as 3 and 5, 5 and 7, and 3, etc. The Twin Prime Conjecture asserts that there are infinitely many twin primes. 7,96 The record twin primes: 6,896,987,339,975 ±
Discrete Mathematics Primes 3-4 Homework Exercises from the Book No., 4,, 5, 6 (page 3)