Erdős problem 243 · problems.science

Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$ .

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243.lean

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Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$ .

Then, for all sufficiently large $n \ge 1$ , $a_n = a_{n-1}^2 - a_{n-1} + 1$ .

FormalConjectures/ErdosProblems/243.leanErdos243.erdos_2434 linesExact file

1∀ (a : ℕ → ℕ),2  StrictMono a →3    Filter.Tendsto (fun n => ↑(a n) / ↑(a (n - 1)) ^ 2) Filter.atTop (nhds 1) →4      (Summable fun x => 1 / ↑(a x)) → ∀ᶠ (n : ℕ) in Filter.atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1

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