Problem 974

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claims/: The 1 claim page of Problem 974, one per claimant's result; the problem's standing derives from them.

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

Statement (precise). 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, if $n$ is odd, $z_1,\ldots,z_n$ are exactly the $n$th roots of unity, and, if $n$ is even, they are the vertices of two regular $(n/2)$-gons with the same circumscribed circle centred at the origin.

Notes. The word "(essentially)" is Erdős's: [Er65b], printed p. 213, display (37), reports the conjecture as Turán's, told to Erdős in conversation, and leaves the word undefined; the site's commentary records that Erdős does not elaborate on what it may mean. Read as the site words it, with the conclusion that the $z_j$ are the $n$th roots of unity, the conjecture fails for every even $n$: $z_1=1$, $z_2=i$ gives $s_k=1+i^k$, which vanishes for every $k\equiv2\pmod4$ (Quanyu Tang, site thread, 20 September 2025), and for $n=2m$ the $m$th roots of unity together with their rotation by $e^{\pi i/(2m)}$ give $s_k=0$ for every $k\not\equiv0,m,3m\pmod{4m}$, a run of $n-1$ zeros in every period of length $4m$ (Tao Hu's construction, posted by Tang the same day). The site's curator, Thomas Bloom, resolves the word through Tijdeman's theorem. The commentary (page last edited 1 October 2025) states the conclusion as "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", credits Tijdeman [Ti66] and labels the problem PROVED (LEAN); replying to the $n=2$ example in the thread on 20 September 2025, Bloom wrote that it "is not a counterexample though", given how vaguely the problem is described; and the formal-conjectures statement erdos_974, which the site's Lean mark follows, concludes that configuration. The precise Statement replaces "(essentially) $z_j=e(j/n)$" by that conclusion and changes nothing else. Under it the problem is proved: Tijdeman [Ti66] proves the classification from two runs of $n-1$ vanishing power sums for pairwise distinct $z_i$, and a single run already forces the $z_i$ to be distinct and nonzero (Proposition 1 of Hu, Tang and Zhang in the thread, proved in both Lean files). Under the site's wording the answer is yes for odd $n$, where the two readings agree, and no for every even $n$, by the construction above, which has no claim page; its authors went on to treat the classification as the problem's resolution. Erdős's setting in [Er65b] also requires $|z_i|\le1$ for $2\le i\le n$, which the site's wording omits; neither answer changes, since the conclusion forces $|z_i|=1$ and the construction lies on the unit circle. Unread: Turán's own statement of the conjecture, and its statement in [Ti66].

Status. PROVED (LEAN). The site's label describes the precise Statement. The site credits Tijdeman [Ti66], who proved the stronger form with two runs of vanishing power sums, and it notes an independent proof in its thread. Its Lean marker refers to third-party formalizations that this corpus has not built. The standing derives from Tijdeman's claim page.

Source. erdosproblems.com/974, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #974, https://www.erdosproblems.com/974.

References.

• [Er65b] Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196-244; printed p. 213, display (37). Library home: erdos_1965_recent_advances_current_problems_number_theory.
• [Ti66] Tijdeman, R., On a conjecture of Turán and Erdős. Indag. Math. (1966), 374-383.

Formalization. Statement in formal-conjectures (at the commit the link pins), whose main theorem erdos_974 concludes Tijdeman's configuration, the site's reading of "essentially", and whose formal_proof attribute points to a third-party Lean proof; that proof and the gist it re-hosts are formalization links on the claim page. The corpus has built neither.

Current assessment

The problem is proved under the precise Statement, by Tijdeman's classification, the accepted claim recorded on Tijdeman's claim page with its refereed publication and the curator's credit, stated as the site and its thread give it. The even-$n$ construction in the Notes answers the site's wording, in the negative for every even $n$, and has no claim page.

The independent proof by Hu, Tang and Zhang (September 2025), given in the thread's comments and in a repository whose manuscript covers odd $n$, and a Lean formalization posted as a gist by Jeremy Tan Jie Rui on 27 April 2026 and Boris Alexeev's re-hosting of it, first committed on 7 May 2026, are disclosed on the claim page; the site credits Tijdeman, and a later proof of a credited result is disclosed on the credited page rather than given a page of its own. Search scope: the site page and its thread, as of 2026-10-07; no wider literature search was made.