Erdős problem 974 · problems.science

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 $e(x)=e^{2\pi ix}$ .

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FormalConjectures/ErdosProblems/

974.lean

Retained formal statement2 of 2

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.

FormalConjectures/ErdosProblems/974.leanErdos974.erdos_974.variants.two_tuples5 linesExact file

1∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ),2  z 0 = 1 →3    ∀ {a b : ℕ},4      a ≠ b →5        (∀ j < n - 1, ∑ i, z i ^ (a + j) = 0) → (∀ j < n - 1, ∑ i, z i ^ (b + j) = 0) → Erdos974.IsTuranConfiguration z

SolvedStatement only, no proof

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