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import FormalConjecturesUtilErdős Problem 243
open Filter
open scoped Topology
namespace Erdos243
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 $n \ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.
@[category research open, AMS 40]
theorem erdos_243 (a : ℕ → ℕ) (ha₀ : StrictMono a)
(ha₁ : Tendsto (fun n ↦ (a n : ℝ) / a (n - 1) ^ 2) atTop (𝓝 1))
(ha₂ : Summable ((1 : ℚ) / a ·)) :
∀ᶠ n in atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1 := a:ℕ → ℕha₀:StrictMono aha₁:Tendsto (fun n => ↑(a n) / ↑(a (n - 1)) ^ 2) atTop (𝓝 1)ha₂:Summable fun x => 1 / ↑(a x)⊢ ∀ᶠ (n : ℕ) in atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1
All goals completed! 🐙
end Erdos243