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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.

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This is a very well-known false conjecture due to Euler.

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    $\begingroup$ ... and that it is false is due to Lander & Parkin ($m=5$, $n=4$) and Elkies ($m=4$, $n=3$). $\endgroup$ – ACL Mar 31 '15 at 12:14
  • $\begingroup$ What about the converse (namely that $x_1^n + \cdots + x_n^n = 1$ has a rational solution for all $n$)? I believe that's an open problem. $\endgroup$ – Adam P. Goucher Apr 2 '15 at 13:47

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