PROVED
This has been solved in the affirmative.
Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of\[s_k=\sum_{1\leq i\leq n}z_i^k\]contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially)\[z_j=e(j/n),\]where $e(x)=e^{2\pi ix}$.
A conjecture of Turán. Erdős speculates that this may be true if there are two distinct $(n-1)$-tuples of consecutive values of $s_k$ which are $0$. He does not elaborate on what the 'essentially' may mean precisely.
This is true (in the stronger form with only two such tuples) - in fact if $n$ is odd then the $z_i$ must be exactly the $n$th roots of unity, and if $n$ is even they must be the vertices of two regular $(n/2)$-gons with the same circumscribed circle centred at the origin. This was first proved by Tijdeman
[Ti66]. An independent proof of this was given in the comments section by Hu, Tang, and Zhang.
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This page was last edited 01 October 2025.
Additional thanks to: Koishi Chan, Tao Hu, and Quanyu Tang
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #974, https://www.erdosproblems.com/974, accessed 2026-01-16
