For each positive integer n, S(n) is defined to be the greatest integer such that, for every positive integer k ≤ S(n), n^2 can be written as the sum of k positive square integers.

(a) Prove that S(n) ≤ n^2 − 14
for each n ≥ 4.

(b) Find an integer n such that
S(n) = n^2 − 14.

(c) Prove that there are infinitely many integers n such that S(n) = n^2 − 14.