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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..

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    Any fourth power is either $0,1 \pmod{16}.$ So any number $n \equiv 15 \pmod {16}$ needs at least $15$ odd fourth powers.2012-10-24

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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.

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    @MargretButton, the Dirichlet thing is in http://en.wikipedia.org/wiki/Whittaker_and_Watson along with chapters on special functions, many analysis topics of use in number theory.2012-10-24