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import FormalConjecturesUtilErdős Problem 463
open Filter
namespace Erdos463
Is there a function $f$ with $f(n)\to\infty$ as $n\to\infty$ such that, for all large $n$, there is a composite number $m$ such that $$ n + f(n) < m < n + p(m) $$ Here $p(m)$ is the least prime factor of $m$.
@[category research open, AMS 11]
theorem erdos_463 : answer(sorry) ↔ ∃ (f : ℕ → ℕ) (_ : Tendsto f atTop atTop),
∀ᶠ n in atTop,
∃ m, m.Composite ∧
n + f n < m ∧ m < n + m.minFac := ⊢ True ↔ ∃ f, ∃ (_ : Tendsto f atTop atTop), ∀ᶠ (n : ℕ) in atTop, ∃ m, m.Composite ∧ n + f n < m ∧ m < n + m.minFac
All goals completed! 🐙
end Erdos463