Chapter 1: The integers and Obviously, this result could not be true - PDF document

The well-ordering axiom The following well-ordering property holds in N : F18PA2 Pure Mathematics A Theorem 1 If S is a non-empty set of positive integers then it Number Theory & Geometry contains an integer m such that m � a for all a ∈ S. Remarks 2 In other words, any non-empty set of positive integers S contains a smallest element. Chapter 1: The integers and Obviously, this result could not be true for an empty set, but when using the well-ordering axiom we have to remember to check that divisibility the sets we are applying it to are in fact non-empty. This harmless assertion turns out to be the principle that underlies much of our work with N and Z . Note. The well-ordering property does not hold in the set of February 19, 2016 rational numbers Q . E.g. the set { q ∈ Q : q > 0 } does not have a smallest element. 1 / 347 5 / 347 The integers Divisibility We will use the set of integers : Let a and b be integers. We say that b is a multiple of a , and write a | b , if there exists an Z = { . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , . . . } integer q such that aq = b . Note that q = b / a . If b is not a multiple of a we write a � | b . and the non-negative integers; Equivalently, we say that a is a divisor , or a factor , of b , or that a N 0 = { 0 , 1 , 2 , 3 , 4 , . . . } divides b , the positive integers We say that a is a proper divisor of b if 1 � a < b . Be careful to distinguish a | b (statement ‘ a divides b ’) from a N = { 1 , 2 , 3 , 4 , . . . } division such as a / b (the number ‘ a divided by b ’). E.g. 4 | 8 (8 / 4 = 2), 4 � | 9 (9 / 4 = 2 . 25). Note. Some people use the notation N for the set { 0 , 1 , 2 , 3 , 4 , . . . } and some people use it for the set { 1 , 2 , 3 , 4 , . . . } . To avoid this ambiguity we will use the above definitions, but we will usually avoid this notation and use the terminology ‘non-negative’ and ‘positive’ integers as defined above. 2 / 347 6 / 347 The integers Divisors The set of integers Z comes equipped with: It is easy to make a table of divisors of positive integers (we need only consider positive (+ve) divisors): ◮ basic rules for arithmetic using addition and multiplication; 1 divides every positive integer ◮ ordering relation � with its rules; 2 divides every second positive integer � 2 ◮ key extra property: the well-ordering of the positive integers. 3 divides every third positive integer � 3 , and so on: n +ve divisors of n n +ve divisors of n 1 1 7 1 7 2 1 2 8 1 2 4 8 3 1 3 9 1 3 9 4 1 2 4 10 1 2 5 10 5 1 5 11 1 11 6 1 2 3 6 12 1 2 3 4 6 12 3 / 347 7 / 347 The relation � Properties of divisibility Proposition 3 For any a , b , c , d ∈ Z : ◮ we will use the order notation � , � , <, > ( a ) a | 0 , but if a � = 0 then 0 � | a , ◮ we write, for example, a � b if the integer a is less than or equal to the integer b , so: ( b ) 1 | a and a | a , ( c ) if a | b and c | d then ac | bd , − 3 � 2 , and 0 � 2 and 2 � 2 . ( d ) if a | b and b | c then a | c , Note that a � b allows for the possibility that a and b are ( e ) if a | b and b | a then a = ± b , equal, whereas a < b asserts that a is strictly less than and ( f ) if a | b and b � = 0 then | a | � | b | , definitely not equal to b . ( g ) a | b if and only if ( − a ) | b , if and only if a | ( − b ) . ◮ any negative number is less than or equal to any positive number: so − 10 000 000 < 1 . Note. These properties should be obvious, but read them carefully ◮ for every pair of integers a , b either a � b or b � a (or both). and make sure you understand why they are true. ◮ if a � b and b � a then a = b . It is impossible for a < b and b < a to both be true. 4 / 347 8 / 347

Some interesting functions Summing over divisors For a positive integer n we define the functions: Note. We get the same values in the sum for F if we write � � ◮ τ ( n ) is the number of positive divisors of n (incl 1 and n ), F ( n ) = f ( d ) = f ( n / d ) . ◮ σ ( n ) is the sum of the positive divisors of n (incl 1 and n ) d | n d | n = n + the sum of the proper divisors of n . We call this the usual trick for summing over divisors (we use this several times below, so make sure you understand it). F (12) = � n +ve divisors τ ( n ) σ ( n ) n +ve divisors τ ( n ) σ ( n ) d | 12 f ( d ) = f (1) + f (2) + f (3) + f (4) + f (6) + f (12) = � 1 1 1 1 7 1 7 2 8 d | 12 f ( n / d ) = f (12) + f (6) + f (4) + f (3) + f (2) + f (1) 2 1 2 2 3 8 1 2 4 8 4 15 By the definitions of τ and σ we have: 3 1 3 2 4 9 1 3 9 3 13 4 1 2 4 3 7 10 1 2 5 10 4 18 Proposition 4 � 5 1 5 2 6 11 1 11 2 12 ◮ τ ( n ) = 1 1( d ) . 6 1 2 3 6 4 12 12 1 2 3 4 6 12 6 28 d | n � � ◮ σ ( n ) = id( d ) = d . d | n d | n 9 / 347 13 / 347 A first look at prime numbers Perfect numbers An integer p > 1 is a prime number if its only divisors are 1 and p . A positive integer n is perfect if σ ( n ) = 2 n . By definition, the number 1 is not a prime number. Or: n is perfect if it equals the sum of its proper divisors. We see that, for a positive integer n , The two smallest perfect numbers are 6 (= 1 + 2 + 3) and 28 (= 1 + 2 + 4 + 7 + 14) n is prime ⇐ ⇒ τ ( n ) = 2 ⇐ ⇒ σ ( n ) = n + 1 . the third is 496 . The first few primes are Our first observation is that these perfect numbers are all even, and this leads us to the next theorem. 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , Theorem 5 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , . . . ⇒ N = 2 n − 1 (2 n − 1) , with 2 n − 1 prime. N is even and perfect ⇐ and we shall see later that the list of primes goes on forever. The ⇐ implication was in Euclid’s Elements (circa 300 BC). The ⇒ implication was proved by Euler (1747) (but not published in his lifetime). 10 / 347 14 / 347 Some dull functions Perfect numbers For a positive integer n we also define the functions: Using Theorem 5, we find the following: 2 n − 1 Is 2 n − 1 prime? N = 2 n − 1 (2 n − 1) ◮ 1 1( n ) = 1 for all n , n ◮ ε (1) = 1 and ε ( n ) = 0 if n > 1 , 1 1 no 1 2 3 yes 6 ◮ id( n ) = n for all n . 3 7 yes 28 These functions seem trivial, but they give us a convenient 4 15 no 120 notation for various operations. 5 31 yes 496 6 63 no 2016 7 127 yes 8128 The next perfect number occurs for n = 13 , when N = 33 550 336 . 11 / 347 15 / 347 Summing over divisors Proof of Theorem 5 Given a function f defined on N , we obtain another function F by We will use the following key property of σ (proof later in course): defining if q is odd then σ (2 m q ) = σ (2 m ) σ ( q ) . � (M) F ( n ) = f ( d ) , n ∈ N . Suppose that N = 2 n − 1 (2 n − 1) , with 2 n − 1 prime. ( ⇐ ) d | n By (M), σ ( N ) = σ (2 n − 1 ) σ (2 n − 1) . That is, the value of F at an integer n is the sum of the values of Since the divisors of 2 n − 1 are 1 , 2 , 4 , . . . , 2 n − 1 , we have f at all the divisors d of n (this is the meaning of the d | n term below the summation sign Σ in the above formula). So, σ (2 n − 1 ) = 1 + 2 + 4 + · · · + 2 n − 1 = 2 n − 1 . (1) F (1) = f (1) Since 2 n − 1 is prime, we have σ (2 n − 1) = 1 + (2 n − 1) = 2 n . F (2) = f (1) + f (2) Hence. F (3) = f (1) + f (3) σ ( N ) = 2 n (2 n − 1) = 2(2 n − 1 (2 n − 1)) = 2 N , F (4) = f (1) + f (2) + f (4) which proves the ⇐ implication. . . . [proof of (1): F (12) = f (1) + f (2) + f (3) + f (4) + f (6) + f (12) S = 1 + 2 + · · · + 2 n − 1 = ⇒ 2 S = 2 + · · · + 2 n − 1 + 2 n , . now subtract and cancel] . . 12 / 347 16 / 347