erdos_680.variant

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

theorem erdos_680.variant : answer(sorry) ↔ ∀ ε > 0, ∃ C > 0,
    ¬ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k ≠ 0,
    Nat.minFac (n + k) > exp ((1 + ε) * √k) + C

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