The question is posed as such:
If
G(k) = min{ g : every "sufficiently large" natural number can be written as the sum of g kth powers }
Then I seek to prove two things. First, to establish the lower bound of : $G(K) \ge k + 1$
and then give a better lower bound for G(4), namely $G(4) \ge 15 $
So for the first, I need to somehow show that for every k, there are numbers that can't be written as the sum of k kth powers. We've typically been working with small k (Proved G(3), and G(4)). So I'm not sure how to generalize to very large values of k.
Next, I've been given a hint by a fellow student that I need to show that every fourth power is congruent to 0 or 1 (mod 16). This fact is easy enough, but not sure what it has to do with anything. Help!!!??
Note: I'm an undergraduate taking a mixed graduate/4th year class. This question is meant for the graduate students only, but I've decided to have a crack at it. To be honest, I'm not even sure I know where to start..