FormalConjectures.ErdosProblems.«419»

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import FormalConjecturesUtil

Erdős Problem 419

References:

erdosproblems.com/419

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[EGIP96] Erdős, Paul and Graham, S. W. and Ivić, Aleksandar and Pomerance, Carl, On the number of divisors of $n!$. (1996), 337--355.


open Filter

open scoped ArithmeticFunction.sigma


namespace Erdos419

The ratio $\sigma_0((n+1)!)/\sigma_0(n!)$, where $\sigma_0$ is the divisor-counting function.

noncomputable def factorialDivisorRatio (n : ℕ) : ℝ :=
  (σ 0 (n + 1).factorial : ℝ) / (σ 0 n.factorial : ℝ)

The set ${1} \cup {1+1/k : k \geq 1}$.

def limitPointSet : Set ℝ :=
  {1} ∪ {x | ∃ k : ℕ, 1 ≤ k ∧ x = 1 + 1 / (k : ℝ)}

If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of $$ \frac{\tau((n+1)!)}{\tau(n!)}? $$

The limit points are exactly ${1} \cup {1+1/k : k \geq 1}$.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos419.lean"]
theorem declaration uses 'sorry'erdos_419 :
    {x : ℝ | MapClusterPt x atTop factorialDivisorRatio} = limitPointSet := by⊢ {x | MapClusterPt x atTop factorialDivisorRatio} = limitPointSet
  sorryAll goals completed! 🐙


end Erdos419