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.
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.