Let S=\{4,8,9,16, \ldots\} be the set of integers of the form m^{k} for integers m, k \geq 2. For a positive integer n, let f(n) denote the number of ways to write n as the sum of (one or more) distinct elements of S. For example, f(5)=0 since there are no ways to express 5 in this fashion, and f(17)=1 since 17=8+9 is the only way to express 17.
(a) Prove that f(30)=0.
(b) Show that f(n) \geq 1 for n \geq 31.
(c) Let T be the set of integers for which f(n)=3. Prove that T is finite and non-empty, and find the largest element of T.
Note that all even elements in S are divisible by 4.