Question

I've asked the same question 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 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!

Answer 1

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.