Euler's false attempted generalization of Fermat's last theorem

Statement

This is the false statement that Euler conjectured:

Suppose $n$ is a natural number greater than $2$. Then, there is no solution to the equation:

$x_{1}^{n}+x_{2}^{n}+\dots +x_{n-1}^{n}=z^{n}$,

with all the $x_{i}$ nonnegative integers and at least two of the $x_{i}$ strictly positive.

In words, no $n^{th}$ power can be expressed as the sum of fewer than $n$ smaller $n^{th}$ powers.

Partial truth

Weaker statements that are true

Further information: Fermat's last theorem for cubes, Fermat's last theorem Although the statement is false in general, it is true for $n=2$. Moreover, Fermat's last theorem, which would be an immediate corollary of this conjecture if it were true, is in fact true.

Analogue for the polynomial ring is true

The analogous statement is true where the $x_{i}$ are not all constant polynomials, in the polynomial ring over a field.

Proof (of falsity)

The statement fails for $n=4$: there exists a fourth power expressible as a sum of three fourth powers.