I've asked the [same question](https://math.stackexchange.com/questions/38238/hilbert-waring-theorem-using-the-sum-of-squares-function) at the Math Stack Exchange site, but I didn't have any luck there. So I'm posting the same question here. 


Denote by $r\_{s,k}(x)$, the number of ways in which $x$ can be expressed as the sum of $k$ $s^{\text{th}}$ powers of integers. Now, the [Hilbert-Waring theorem](http://en.wikipedia.org/wiki/Waring%27s_problem) is equivalent to the following.
$$\forall s\in \mathbb{N}\; \exists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N}$$
 Now, call me optimistic, but I've been trying to prove this fact via contradiction. So, assuming the contradiction gives us the following.
$$\forall s\in \mathbb{N}\; \nexists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N}$$
 Now, if this is true, then it means that there is *at least one* $s$ for which there exists a corresponding $k$, such that $r\_{s,k}(x)\geq 1$. So, for all the other integers $s$, the opposite must hold.


I have come this far till now. If we can show somehow that from the above arguments it must follow that $r\_{s,k}(x)=0$ the proof will follow, since for any natural $a$, $r\_{s,k}(a^s)$ is always greater than or equal to one. 


I'd be grateful to anyone who pitches in any ideas on how to approach this problem, or if it's hopeless to do so. :) 


Many thanks in advance!



The Hilbert-Waring theorem using the sum-of-squares function.
The negation of the statement
$$ \forall s\in \mathbb{N}\; \exists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N} $$
is not the statement
$$ \forall s\in \mathbb{N}\; \nexists\: k \; \: \text{such that} \;r\_{s,k}(x)\geq 1\; \forall\; x \in \mathbb{N} $$
but the statement
$$ \exists s\in \mathbb{N}\; \forall\: k \; \exists x \in \mathbb{N}\: \text{such that} \;r\_{s,k}(x)=0 $$
In other words: if the Hilbert-Waring theorem is false, then there is some exponent $s$ such that for any number of summands $k$ there exists a natural number $x$ which is not the sum of $k$ $s$-th powers. Your confusion comes from the fact that you made a basic error in logic.


**EDIT 1:** Perhaps I should add that the Hilbert-Waring theorem is far from trivial, all known proofs are quite complicated.


**EDIT 2:** I should also add that in the theory of the Waring problem one usually denotes by $k$ the exponent and by $s$ the number of summands, not the other way.


