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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.