Fermat's Last Theorem in multiple variables I was wondering if there was anything we could say about when, given $m$,
$\exists n (\forall x_1,\dots,x_m \in \mathbb{N} ( x_1^n + x_2^n + \dots + x_{m-1}^n \neq x_m^n))$
Fermat's Last Theorem implies this for $m=3$.
The follow up question would be whether there is an algorithm to find such an $n$ for a given $m$.  The question came out of a question in a computer science class. In truth, I'm looking for an integral polynomial that, given some $m$, there is exactly 1 integral solution to $p(x_1, \dots, x_m)$ with the $x_i < m$.
 A: If you allow your variables $x_i$ to be non-negative integers, then this is related to Waring's problem. Let $g(n)$ be the minimum number $s$ such that every natural number can be represented as a sum of at most $s$ natural numbers to the $n$th power. If $m-1\geq g(n)$, then given any $x_m\in \mathbb{N}$, the number $x_m^n$ can be represented as the sum of $x_1^n+x_2^n+\cdots +x_{m-1}^n$, for some non-negative integers $x_i\geq 0$.
Hence, if we fix $m$, and there is an $n$ such that $x_1^n+\cdots+x_{m-1}^n=x_m^n$ has no non-negative solutions, then we obtain a bound of $m-1<g(n)$, which gives a lower bound for $n$. The function $g(n)$ is conjectured to be given by
$$g(n) = 2^n + [(3/2)^n] − 2$$
where $[\cdot]$ is the greatest integer function. The first few values (of the conjectural formula for $g(n)$) are:
$$g(1) = 1,\ g(2)= 4,\ g(3)= 9,\ g(4)= 19,\ g(5)= 37,\ g(6)= 73,\ g(7)= 143,\ g(8)= 279,\ldots$$
So, for instance, if $m=3$, this would say that $n\geq 2$ (and Fermat's last theorem says that $n=3$ works). If $m=4$, then $n\geq 2$. If $m=5,6,7,8,$ or $9$, then $n\geq 3$, etc.
