Problem 266

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claims/: The 1 claim page of Problem 266, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t\geq 1$ such that

$$\sum \frac{1}{a_n+t}$$

is irrational.

Status. Disproved: Kovač and Tao's 2024 construction of a sequence whose shifted reciprocal sums are rational for every rational shift is recorded on its claim page. The site (page last edited 2025-09-28) labels the problem DISPROVED (LEAN) and credits them with the negative answer; the Lean proof behind the label is a public formalization of their argument linked from the claim page, not built or audited in this corpus.

Source. erdosproblems.com/266, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #266, https://www.erdosproblems.com/266.

References.

• [KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

Formalization. Statement in formal-conjectures at the linked commit, tagged research solved, whose formal_proof attribute cites a Lean 4 proof in the public lean-proofs repository at a commit of 2026-09-15; that proof is linked from the claim page and is not built or audited in this corpus.

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Linked library material

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• kovac_2024_several_irrationality_problems_ahmes_series