One of the great discoveries of the pythagorean school is that the side and diagonal of a square are not commensurable. namely in modern parlance, the ratio of the length of the diagonal to the length of the side of a square is not a rational number. as an example, this means that the square root of 2 is not rational.
if m is a positive integer, explain why each prime in the factorization of m2 must occur an even number of times.
suppose the square root of 2 is rational. so square root of 2 equals a, b ∈ z . conclude that a2 = 2b22q==. from this, determine the number of times that the prime 2 occurs on each side of the equation, and derive a contradiction to the unique factorization theorem (every integer can be uniquely factored into a product of primes, up to ordering).