Bounds for Waring's Problem 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..
 A: I did $G(4)$ in a comment. The bit about $$ G(k) \geq k+1$$ is actually about $k$-dimensional volume. Define a constant $C_k$ to be the volume of the set
$$ x_1, x_2, \ldots, x_k \geq 0, x_1^k + x_2^k + \cdots x_k^k \leq 1.  $$ The important thing is that $0 < C_k < 1,$ as we are describing part of the standard unit cube.
By a technique due to Dirichlet,
$$  C_k = \left( \Gamma \left(  1 + \frac{1}{k} \right) \right)^k   $$
Now, for large integer $N,$ the number of integer lattice points satisfying 
$$  x_1, x_2, \ldots, x_k \geq 0, x_1^k + x_2^k + \cdots x_k^k \leq N    $$ is well approximated by $C_k N.$ So, the number of possible values represented integrally by 
$$  x_1^k + x_2^k + \cdots x_k^k  $$  is no larger than $C_k N.$ So, no matter how large $N$ is, there are roughly $(1 - C_k) N$ numbers up to $N$ tha are not represented. In particular the nonrepresented numbers are arbitrarily large.
Very similar: there are infinitely many positive integers that are not represented integrally by
$$  x^2 + y^3 + z^6, \; y \geq 0. $$ 
EDIT: From the viewpoint of counting lattice points, the surprise is that $ x^2 + y^2 + z^9 $ does not represent all sufficiently large numbers, in that $$ x^2 + y^2 + z^9 \neq 216 p^3 $$ for (positive) prime $p \equiv 1 \pmod 4$ and integers $x,y,z,$ even if we are generous and allow $z$ negative. This proof is quite easy.
See if you can borrow the second edition (1997) of The Hardy-Littlewood Method by R. C. Vaughan. I'm in it. 
