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 $\mathbb{N}$.



On Generalizations of Fermat's Conjecture
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](http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture) so you are in good company, and $n \ge 6$ is still open. See that page for further details. 


