Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not.


The equation $x\_1^m+\cdots+x\_n^m=1$ has nonzero rational solutions iff $n\geq m$.


Here a nonzero rational solution means nonzero $y\_1,\cdots,y\_n\in\mathbb{Q}$ satisfying the above equation.


When $n=2$, the above conjecture is confirmed by Fermat's last theorem.



A stronger version of Fermat's last theorem
This is a very well-known false conjecture [due to Euler](http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture).


