FormalConjectures.ErdosProblems.«681»

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import FormalConjecturesUtil

Erdős Problem 681

Reference: erdosproblems.com/681


namespace Erdos681

IsLPF p m says that p is the least prime factor of m.

def IsLPF (p m : ℕ) : Prop := p.Prime ∧ p ∣ m ∧ ∀ q, q.Prime ∧ q ∣ m → p ≤ q

Erdős problem 681. Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$, where $p(m)$ is the least prime factor of $m$ ?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_681 : answer(sorry) ↔
    ∀ᶠ n in .atTop, ∃ k > 0, (n + k).Composite ∧ ∀ p, IsLPF p (n + k) → p > k ^ 2 := by⊢ True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k > 0, (n + k).Composite ∧ ∀ (p : ℕ), IsLPF p (n + k) → p > k ^ 2
  sorryAll goals completed! 🐙


end Erdos681