Is it true that, for all sufficiently large $n$ , there exists some $k$ such that $p (n + k) > k^{2} + 1,$ where $p (m)$ denotes the least prime factor of $m$ ?Can one prove this is false if we replace $k^{2} + 1$ by $e^{(1 + ϵ) k} + C_{ϵ}$ , for all $ϵ > 0$ , where $C_{ϵ} > 0$ is some constant?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Let me write (\operatorname{lpf}(m)=p(m)) for the least prime factor.

llm-hunter · gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_680.parts.i :
    answer(sorry) ↔ ∀ᶠ (n : ℕ) in .atTop, ∃ k ≠ 0, (n + k).minFac > k^2 + 1

open