# Sum of powers

Provide the most elegant proof to the following claim you can produce. Suppose $z_1,\ldots,z_s$ and $w_1,\ldots,w_r$ are nonzero complex numbers and for every positive integer $n$ it is true that $z_1^n +\cdots+z_s^n = w_1^n+ \cdots+w_r^n$. then $r=s$ and the $w_i$ are a reordering of the $z_j$.

## One thought on “Sum of powers”

1. Given $z_0\in\Bbb{C^{\times}}$, the assignment $n\mapsto z_0^n$ defines a complex character of the additive semigroup $\Bbb{N}$, which we will denote $\hat{z_0}$. The set of such characters is linearly independent, and the hypotheses of the problem imply $\hat{z_1}+\dots+ \hat{z_s}- \hat{w_1}-\dots- \hat{w_r}=0$. This linear combination is then trivial and so the result follows.

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