Number Theory and Methods of Proof

1. Number Theory and Methods of Proof • Content:Properties of integer, rational and real numbers. • Underlying theme:Methods of mathematical proofs.

2. This Lecture • Even and odd integers • Prime and composite integers • Constructive and nonconstructive proofs • Method of direct proof

3. Even and Odd Integers • Definition: An integer n ● is even iff  an integer k such that n=2k; ● is odd iff  an integer k such that n=2k+1. Ex: If x and y are integers, is even or odd?

4. Prime and Composite Integers • Definition: An integer n (which is >1) ● is primeiff positive integers r and s, if n=r·s then r=1 or s=1; ● is compositeiff positive integers r and s such that n=r·s and r≠1 and s≠1. • Examples: 5, 7, 23 are prime; 4, 22, 16042 are composite.

5. Proving Existential Statements • Existential statement: xD such that P(x) • 2 proof methods for existential statements: ● Constructive proofs; ● Nonconstructive proofs.

6. Constructive Proofs of Existence • 2 ways: 1) Find x that makes P(x) true. 2) Give a set of directions for finding such x. • Example: 1) There are integer numbers a,b and c such that Proof: For a=3, b=4 and c=5,

7. Constructive Proofs of Existence 2)Suppose a,b Z such that 1<a<b. Prove that there is a composite even integer c such that a2<c<b2 . Proof: By division into cases: (a) Suppose a is even. Then a=2k for some integer k. (by definition) Hence c=a·b = (2k)·b = 2·(k·b) is even integer; (because k·b is an integer) c=a·b is composite (because a≠1 and b≠1); c=a·b>a2(because a<b). c=a·b<b2(because a<b). (b) Suppose b is even. (c) Suppose both a and b are odd.

8. Nonconstructive Proofs of Existence • 2 ways: (1) Show that the existence of x is guaranteed by an axiom or apreviously proved theorem. (2) Show that the assumption that there is no such x leads to a contradiction. • Disadvantage: Often these methods give no clue how to find x.

9. Proving Universal Statements • Universal statement: xD if P(x) then Q(x) • Proof methods for universal statements: ● Method of exhaustion; ● Method of generalizing from the generic particular. (show the property for a particular but arbitrarily chosen x)

10. Method of Direct Proof • The statement: xD if P(x) then Q(x). • Suppose x is a particular but arbitrarily chosen element of D for which P(x) is true; • Show the conclusion Q(x) is true by using ♦ definitions; ♦ previously established results; ♦ rules of logical inference.

11. Method of Direct Proof (Ex.) • Show xZifx is odd then3x+9 is even. Proof: Suppose x is an arbitrarily chosen odd integer. Then x=2k+1 for some integer k. (by definition) So 3x+9 = 3(2k+1)+9 (by substitution) = 6k+3+9 (by distributive law) = 2(3k+6) (by factoring out a 2)(*) 3k+6 is an integer. (**) Hence 3x+9 is even based on (*), (**) and definition of even integers. ▀ (this is what we needed to show)

12. Directions for writing proofs • Write the theorem to be proved. • Clearly mark the beginning of your proof with the word Proof. 3) Make your proof self-contained. (Identify all variables used in the proof; state the sources of outside facts). 4) Write proofs in complete English sentences.