For integer $n \geq 1$ we define the factor difference set of $n$ by $D (n) = {∣ a - b ∣ : n = ab} .$ Is it true that, for every $k \geq 1$ , there exist integers $N_{1} < \dots < N_{k}$ such that $∣ \cap_{i} D (N_{i})∣ \geq k ?$

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 'difference/covering' (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

The statement you wrote is **Erdős problem #885** (in the formulation used on the Erdős Problems site), and what is currently known is:

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_885 : answer(sorry) ↔ ∀ k ≥ 1,
    ∃ Ns : Finset ℕ,
      (∀ n ∈ Ns, 1 ≤ n) ∧
      Ns.card = k ∧
      (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k

formal-conjectures/885.lean ↗

status

open

evidence

formal

status

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