TITLE: On Generalizations of Fermat's Conjecture
QUESTION [12 upvotes]: We know the following facts:
(1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$.
(2) For all $3\leq n$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has no solution in $\mathbb{N}$.
Question: Is the following generalization true?
For all $2\leq m$ both of the following statements are true:
(1) For all $1\leq n\leq m$ the equation $x_{1}^{n}+...+x_{m}^{n}=x_{m+1}^{n}$ has a solution in $\mathbb{N}$.
(2) For all $m+1\leq n$ the equation $x_{1}^{n}+...+x_{m}^{n}=x_{m+1}^{n}$ has no solution in $\mathbb{N}$.

REPLY [25 votes]: No this is not true. For $n=4$ one has 
$$2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$$
found by Elkies (as part of an infinite family of solutions).
Also earlier  it was known, for example, for $n=5$, 
$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$
by Lander and Parkin. 
But part 2 was a conjecture of Euler so you are in good company, and $n \ge 6$ is still open. See that page for further details.

REPLY [10 votes]: The smallest counterexample for $n = 4$ is
$95800^4 + 217519^4 + 414560^4 = 422481^4$.
This has been found out by Roger Frye in 1988, cf.
http://euler.free.fr/docs/euler88.ps.