FormalConjectures.ErdosProblems.«379»

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import FormalConjecturesUtil

Erdős Problem 379

Reference: erdosproblems.com/379


namespace Erdos379


open Filter


noncomputable def S (n : ℕ) : ℕ :=
  sSup {s | ∀ k ∈ Finset.Ico 1 n, ∃ p, p.Prime ∧ p^s ∣n.choose k}

Let $S(n)$ denote the largest integer such that, for all $1 ≤ k < n$, the binomial coefficient $\binom{n}{k}$ is divisible by $p^S(n)$ for some prime $p$ (depending on $k$).Then $\limsup S(n) = \infty$.

This was formalized in Lean by Tao.

@[category research solved, AMS 11,
formal_proof using lean4 at "https://github.com/teorth/analysis/blob/4f623b0f4cacdb967f1f8132db0becaee0f1fb3d/Analysis/Misc/erdos_379.lean#L90",
formal_proof using formal_conjectures at "https://github.com/XC0R/formal-conjectures/blob/80a965e9a85d3f3dabd0a398a49adab6742ea6e0/FormalConjectures/ErdosProblems/379.lean#L123"]
theorem declaration uses 'sorry'erdos_379 : atTop.limsup (fun n => (S n : ℕ∞)) = ⊤ := by⊢ limsup (fun n => ↑(S n)) atTop = ⊤
  sorryAll goals completed! 🐙


end Erdos379