Math 3000 - Lectures 5 & 6

Contradiction Proofs

1. There are an infinite number of primes.

2. There do not exist three consecutive natural numbers such that the cube of the largest is the sum of the cubes of the smaller numbers.

Pf. Look at (n+2)3 = (n+1)3 + n3
n3 + 6n2 + 12n + 8 = n3 + 3n2 + 3n + 1 + n3
0 = n3 - 3n2 - 9n - 7 = f(n)
f'(n) = 3n2 - 6n - 9 = 3(n-3)(n+1)
the only root of f(n) is non-integral (between 5 and 6).

3. There do not exist prime numbers a,b,c such that c3 = a3 + b3.

Pf. At least one of a, b is even.
May assume b = 2.
8 = c3 - a3 = (c - a)(c2 + ca + a2)
but each term in the sum is greater than or equal to 4 .

4. If a,b,c are integers such that a2 + b2 = c2, show that at least one of a or b is even.

Case Method of Proof

Prove that every natural number n is either a prime, a perfect square or divides (n-1)!

Pigeon-Hole Principle : If kn + 1 objects are distributed among n sets, one of the sets must contain at least k + 1 objects.

Proof by contradiction.

Let there be 9 points in 3-space with integer coordinates. Show that there is a pair of these points whose line segment contains an interior point whose coordinates are integers.