Erdős problem 266 · problems.science

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

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FormalConjectures/ErdosProblems/

266.lean

Retained formal statement2 of 2

In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing sequence $a_n$ of positive integers such that $\sum \frac{1}{a_n + t}$ converges to a rational number for all $t \in \mathbb{Q}$ such that $t \ne -a_n$ for any $n$ .

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. [arXiv:2406.17593](https://arxiv.org/abs/2406.17593) (2024).

FormalConjectures/ErdosProblems/266.leanErdos266.erdos_266.variants.all_rationals1 lineExact file

1∃ a, StrictMono a ∧ a 0 ≥ 1 ∧ ∀ (t : ℚ), (¬∃ n, t = -↑(a n)) → ∃ q, HasSum (fun n => 1 / (↑(a n) + ↑t)) ↑q

SolvedStatement only, no proof

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