# L05_proofs_by_contradiction_print - Proof by Smallest...

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1 Proof by Smallest Counterexample Definitions: log 2 ( n ) is x such that 2 x = n . log 2 ( n ) is the unique i s.t. 2 i n < 2 i +1 Prime factorization of n is the representation of n as multiplication of a list of primes . e.g. 12 = 2 × 2 × 3 , 6! = 2 × 2 × 2 × 2 × 3 × 3 × 5 Define SIZE ( n ) to be the number of prime factors in prime factorization of n . e.g. SIZE (12) = 3 , SIZE (6!) = SIZE (720) = 7 e.g. log 2 (2) = 1 , log 2 (3) = 1 , log 2 (4) = 2 log 2 (31) = 4 , log 2 (32) = 5 , log 2 (33) = 5

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2 Proof by Smallest Counterexample Theorem: For any positive integer n , SIZE ( n ) log 2 ( n ) . Proof: Let P ( n ) be the statement SIZE ( n ) log 2 ( n ) . Assume the theorem is wrong. i.e. There is a smallest integer m s.t. P ( m ) is false . Let p be a prime factor of m . Then,
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