Basic Number Theory The integers are the natural numbers, 0 and the - PowerPoint PPT Presentation

Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide01.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide02.html Basic Number Theory prev | slides | next prev | slides | next Numbers The natural numbers are 1, 2, 3, ... The whole numbers are 0, 1, 2, 3, ... Basic Number Theory The integers are the natural numbers, 0 and the additive inverses of the natural numbers: ..., -3, -2, -1, 0, 1, 2, 3, ... The natural and whole numbers are closed under addition (the sum of any two natural numbers is a natural number) and multiplication. The integers are closed under addition, subtraction and multiplication, but not division. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:36 PM 1 of 1 09/30/2003 08:36 PM Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide03.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide04.html Basic Number Theory Basic Number Theory prev | slides | next prev | slides | next Integers and Division Prime and Composite Numbers If a and b are integers with a 0, then " a divides b " if there is an A natural number p 1 is a prime number if the only numbers that integer c such that b = ac . In this case a is a factor of b and b is a divide it are 1 and p . multiple of a . A natural number is a composite number if it is not prime. We use the notation a | b to say " a divides b ". The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Theorem: Let a , b and c be integers. Then Notice that 2 is the only even prime number; all other even numbers are composite. 1. if a | b and a | c then a | ( b + c ). 2. if a | b then a | bc for all integers c . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 3. if a | b and b | c then a | c . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:36 PM 1 of 1 09/30/2003 08:36 PM

Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide05.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide06.html Basic Number Theory Basic Number Theory prev | slides | next prev | slides | next Fundamental Theorem of Arithmetic Prime Factorization Every natural number can be writen uniquely as the product of Question: How does one find the prime factorization of a number primes, where the prime factors are written in order of increasing like 1820? size. Answer: Attempt division by prime numbers, starting with the For example: smallest. If division is possible continue with the quotient and the same prime number. If division is not possible then move the next 28 = 2 * 2 * 7 largest prime number. 451 = 11 * 41 42 = 2 * 3 * 7 On the next slide we see that 1820 can be factored Question: How does one find the prime factorization of a number 1820 = 2 * 2 * 5 * 7 * 13. like 1820? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:36 PM 1 of 1 09/30/2003 08:36 PM Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide07.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide08.html Basic Number Theory Basic Number Theory prev | slides | next prev | slides | next Prime Factorization Prime Factorization Theorem: If n is a composite natural number then n has a prime Division Possible? Operation Output divisor less than or equal to the square root of n . 2 | 1820 yes 910 = 1820/2 2 2 | 910 yes 455 = 910/2 2 Proof: Suppose that n = ab for nonzero numbers a and b . If a is composite then its prime factors are clearly less than a ; similarly for 2 | 455 no next prime: 3 b . Let’s suppose that both a > sqrt( n ) and b > sqrt( n ) and see if this 3 | 455 no next prime: 5 leads to a contradiction: 5 | 455 yes 91 = 455/5 5 5 | 91 no next prime: 7 ab > sqrt( n ) 2 = n 7 | 91 yes 13 = 91/7 7 which contradicts the fact that n = ab . This means that either a or b 7 | 13 no next prime: 11 is less than the square root of n , which in turn means that n has a 11 | 13 no next prime: 13 prime factor less than its square root. This finishes the proof. 13 | 13 yes 1 = 13/13 13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:36 PM 1 of 1 09/30/2003 08:37 PM

Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide09.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide10.html Basic Number Theory Basic Number Theory prev | slides | next prev | slides | next Prime Factorization Division One consequence of this theorem is that when forming the prime The Division Theorem: Let a be an integer and d be a positive factorization of a number n , we need only consider prime numbers integer. Then there are unique integers q and r with 0 <= r <= d less than or equal to the square root of n . such that For example, if you search for the prime factors of 101, you need a = dq + r . only try dividing 101 by primes up to sqrt(101); the largest of these is 7. Since 2, 3, 5, and 7 are not factors of 101, we can safely d is the divisor , conclude that 101 is prime. a is the dividend , 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 q is the quotient , r is the remainder . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:37 PM 1 of 1 09/30/2003 08:37 PM Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide11.html Basic Number Theory http://localhost/~senning/courses/ma229/slides/number-theory/slide12.html Basic Number Theory Basic Number Theory prev | slides | next prev | slides | next Greatest Common Divisor (GCD) Euclidean Algorithm for the GCD Let a and b be integers not both zero. The largest integer d such that The Euclidean algorithm hinges on the following lemma (recall that d | a and d | b is called the greatest common divisor of a and b and a lemma is a "little theorem"): is denoted gcd( a , b ). Lemma: Let a = bq + r where a , b , q and r are integers. Then The integers a and b are said to be relatively prime if gcd( a , b ) = 1. gcd( a , b ) = gcd( b , r ). Finding the GCD of two integers is easy if their prime Proof: Suppose d divides both a and b . Then d also divides a - bq = factorizations are known. When they are not known the Euclidean r and so d divides both b and r . Algorithm for finding the GCD can be used. Now suppose that d divdes b and r . Then d also divides bq + r = a and so d divides both a and b . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Since all divisors of a and b are also divisors of b and r (and vice versa), we conclude that gcd( a , b ) = gcd( b , r ). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 of 1 09/30/2003 08:37 PM 1 of 1 09/30/2003 08:37 PM