Direct Proof and Counterexample III Lecture 14 Section 3.3 Thu, Feb 3, 2005

Divisibility Definition: An integer a divides an integer b if a  0 and there exists an integer c such that ac = b. Write a | b to indicate that a divides b. Divisibility is a positive property.

Prime Numbers Definition: An integer p is prime if p  2 and the only positive divisors of p are 1 and p. A prime number factors only in a trivial way: p = 1  p. Prime numbers: 2, 3, 5, 7, 11, … Is this a positive property? Is there a positive characterization of primes?

Composite Numbers Definition: An integer n is composite if there exist integers a and b such that 1 < a < n and 1 < b < n and n = ab. A composite number factors in a non-trivial way : n = a  b, a > 1, b > 1. Composite numbers: 4, 6, 8, 9, 10, 12, … Is this a positive property?

Units and Zero Definition: An integer u is a unit if u | 1. The only units are 1 and –1. Definition: 0 is zero.

Example: Direct Proof Theorem: If a | b and b | c, then a | c. Proof: Let a, b, c be integers and assume a | b and b | c. Since a | b, there exists an integer r such that ar = b. Since b | c, there exists an integer s such that bs = c. Therefore, a(rs) = (ar)s = bs = c. So a | c.

Example: Direct Proof Theorem: Let a and b be integers. If a | b and b | a, then a = b. Proof: Let a and b be integers. Suppose a | b and b | a. There exist integers c and d such that ac = b and bd = a. Therefore, acd = bd = a.

Example: Direct Proof Therefore, cd = 1. Thus, c = d = 1 or c = d = -1. Then a = b or a = -b.

Example: Direct Proof Corollary: If a, b  N and a | b and b | a, then a = b. This is analogous to the set-theoretic statement that if A  B and B  A, then A = B. Preview: This property is called antisymmetry. If a ~ b and b ~ a, then a = b.

Example: Direct Proof Theorem: Let a, b, c be integers. If a | b and b | a + c, then a | c. Proof: Let a, b, and c be integers. Suppose a | b and b | a + c. There exist integers r and s such that ar = b and bs = a + c.

Example: Direct Proof Substitute ar for b in the 2nd equation to get (ar)s = a + c. Rearrange the terms and factor to get a(rs – 1) = c. Therefore, a | c.

Example: Direct Proof Theorem: If n is odd, then 8 | (n2 – 1). Proof: Let n be an odd integer. Then n = 2k + 1 for some integer k. So n2 – 1 = (2k + 1)2 – 1 = 4k2 + 4k = 4k(k + 1).

Example: Direct Proof Either k or k + 1 is even. Therefore, k(k + 1) is a multiple of 2. Therefore, n2 – 1 is a multiple of 8. Can you think of an alternate, simpler proof, based on the factorization of n2 – 1?

Example: Direct Proof Theorem: If n is odd, then 24 | (n3 – n). Proof: ?

Example: If-and-Only-If Proof Theorem: Let a, b be integers. Then a | b if and only if a2 | b2. Proof: () Suppose that a | b. Then ac = b for some c  Z. So, a2c2 = b2 and therefore, a2 | b2.

Example: If-and-Only-If Proof Suppose that a2 | b2. Then a2d = b2 for some d  Z. Now what?

The Fundamental Theorem of Arithmetic Theorem: Let n be a positive integer. Then n = p1a1p2a2…pkak, where each pi is prime and each ai is nonnegative. Furthermore, the primes and their exponents are unique.

The Fundamental Theorem of Arithmetic Use the Fundamental Theorem of Arithmetic to complete the previous proof.