Source labels openChecked July 26, 2026

Erdős Problem 243

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 $\sum \frac{1}{a_n} \in \mathbb{Q}$ .

Sequences and series AMS 40

Mathematical statement

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 $n \ge 1$ , $a_n = a_{n-1}^2 - a_{n-1} + 1$ .

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_243

Canonical source

Complete statement target, proof intentionally absentLean 4

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 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/243background