Problem 427

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claims/: The 1 claim page of Problem 427, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that, for every $n$ and $d$, there exists $k$ such that

$$d \mid p_{n+1}+\cdots+p_{n+k},$$

where $p_r$ denotes the $r$th prime?

Status. PROVED (LEAN), on the site's label, which credits Cédric Pilatte's observation that the answer follows from Shiu's theorem on runs of consecutive primes in one residue class [Sh00], and records Lean proofs of the statement outside this corpus; the deduction, the credit and the Lean developments are on the claim page. None of the Lean proofs is built here.

Source. erdosproblems.com/427, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #427, https://www.erdosproblems.com/427.

References.

• [Sh00] Shiu, D. K. L., Strings of congruent primes. J. London Math. Soc. (2) 61 (2000), no. 2, 359-373.

Formalization. Statement in formal-conjectures, at the linked commit of 19 September 2026, whose formal_proof attribute points to the proof in the Jayyhk/erdos-lean repository linked from the claim page, beside the gist and the lean-proofs file it vendors; none is built or audited here.

Progress

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Known Results

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