A) Prove the following theorem by considering two distinct cases. For any integer n, n^{2} + n is even.
B) If x = r^{2 } - s^{2} and y = 2rs for any integers r and s, then x^{2} + y^{2 }= z^{2} for some nonnegative integer z. State the hypothesis of this sentence and the conclusion. Give a direct proof of the statement