1 In this lecture  Number Theory ● Rational numbers ● Divisibility  Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample

2 Common mistakes in proofs Arguing from examples Using same letter to mean two different things Jumping to a conclusion (without adequate reasons)

3 Disproof by counterexample  To disprove statement of the form “  x  D if P(x) then Q(x)”, find a value of x for which ● P(x) is true and ● Q(x) is false.  Ex: For any prime number a, a 2 -1 is even integer. Counterexample: a=2.

4 Rational Numbers  Definition: r is rational iff  integers a and b such that r=a/b and b≠0.  Examples: 5/6, -178/123, 36, 0, …  Theorem: Every integer is a rational number.

5 Properties of Rational Numbers Theorem: The sum of two rational numbers is rational. Proof: Suppose r and s are rational numbers. Then r=a/b and s=c/d for some integers a,b,c,d s.t. b≠0, d≠0. (by definition) So (by substitution) (by basic algebra) Let p=ad+bc and q=bd. Then r+s=p/q where p,q  Z and q≠0. Thus, r+s is rational by definition. ■

6 Types of Mathematical Statements  Theorems: Very important statements that have many and varied consequences.  Propositions: Less important and consequential.  Corollaries: The truth can be deduced almost immediately from other statements.  Lemmas: Don’t have much intrinsic interest but help to prove other theorems.

7 Divisibility Definition: For n,d  Z and d≠0 we say that n is divisible by d iff n=d·k for some k  Z. Alternative ways to say: n is a multiple of d, d is a factor of n, d is a divisor of n, d divides n. Notation: d | n. Examples: 6|48, 5|5, -4|8, 7|0, 1|9.

8 Properties of Divisibility For  x  Z, 1|x. For  x  Z s.t. x≠0, x|0. An integer x>1 is prime iff its only positive divisors are 1 and x. For  a,b,c  Z, if a|b and a|c then a|(b+c). Transitivity: For  a,b,c  Z, if a|b and b|c then a|c.

9 Divisibility by a prime  Theorem: Any integer n>1 is divisible by a prime number.  Sketch of proof: Division into cases: ● If n is prime then we are done (since n | n). ● If n is composite then n=r 1 ·s 1 where r 1,s 1  Z and 1<r 1 <n,1<s 1 <n. (by definition of composite number) (Further) division into cases: ♦If r 1 is prime then we are done (since r 1 |n). ♦ If r 1 is composite then r 1 =r 2 ·s 2 where r 2,s 2  Z and 1<r 2 <r 1,1<s 2 <r 1.

10 Divisibility by a prime  Sketch of proof (cont.): Since r 1 |n and r 2 |r 1 then r 2 |n (by transitivity). Continuing the division into cases, we will get a sequence of integers r 1, r 2, r 3,…, r k such that 1< r k < r k-1 <…< r 2 < r 1 <n ; r p |n for each p=1,2,…,k ; r k is prime. Thus, r k is a prime that divides n. ■

11 Unique Factorization Theorem Theorem: For  integer n>1,  positive integer k, distinct prime numbers, positive integers s.t., and this factorization is unique. Example: 72,000 =