Let $A \subseteq N$ be a set such that $∣ A \cap [1, x]∣ = o (x^{1/2})$ . Let $B = {n \geq 1 : a ∤ n for all a \in A} .$ If $B = {b_{1} < b_{2} < \dots}$ then is it true that $lim \frac{1}{x} b_{i} < x \sum (b_{i + 1} - b_{i})^{2}$ exists (and is finite)?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=none) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'none'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

In full generality, this is **not currently known**. It is recorded as an **open Erdős problem** (Erdős Problems #489): given (A\subseteq\mathbb N) with (|A\cap[1,x]|=o(x^{1/2})) and (B={n\ge1:\ a\nmid n\ \forall a\in A}), it is **open** whether the limit $ \lim_{x\to\infty}\frac1x\sum_{b_i<x}(b_{i+1}-b_i)^2 $ always e…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_489 : answer(sorry) ↔
    ∀ (A : Set ℕ),
      (fun x : ℕ => (((Finset.Icc 1 x).filter (· ∈ A)).card : ℝ)) =o[atTop]
        (fun x : ℕ => (x : ℝ).sqrt) →
      (sievedSet A).Infinite →
      ∃ L : ℝ, Tendsto (fun x : ℕ => GapSumSq A x / (x : ℝ)) atTop (𝓝 L)

formal-conjectures/489.lean ↗

status

open

evidence

formal

status

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