FormalConjectures.ErdosProblems.«392»

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import FormalConjecturesUtil

Erdős Problem 392

Reference: erdosproblems.com/392


open Filter


open scoped Nat


namespace Erdos392

Let $A(n)$ denote the least value of $t$ such that $$ n! = a_1 \cdots a_t $$ with $a_1 \leq \cdots \leq a_t\leq n^2$. Then $$ A(n) = \frac{n}{2} - \frac{n}{2\log n} + o\left(\frac{n}{\log n}\right). $$

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/blob/main/PrimeNumberTheoremAnd/Erdos392.lean"]
theorem declaration uses 'sorry'erdos_392 (A : ℕ → ℕ) (h : ∀ n > 0,
    IsLeast { t + 1 | (t) (_ : ∃ a : Fin (t + 1) → ℕ, (n)! = ∏ i, a i ∧
      Monotone a ∧ a (Fin.last t) ≤ n ^ 2) } (A n)) :
    ((fun (n : ℕ) => (A n - n / 2 + n / (2 * Real.log n) : ℝ)) =o[atTop] fun n => n / Real.log n)
  := byA:ℕ → ℕh:∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n ! = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)⊢ (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[atTop] fun n => ↑n / Real.log ↑n
  sorryAll goals completed! 🐙

If we change the condition to $a_t \leq n$ it can be shown that $$ A(n) = n - \frac{n}{\log n} + o\left(\frac{n}{\log n}\right) $$

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_392.variants.lower (A : ℕ → ℕ)
    (hA : ∀ n > 0, IsLeast
      { t + 1 | (t) (_ : ∃ a : Fin (t + 1) → ℕ, (n)! = ∏ i, a i ∧
        Monotone a ∧ a (Fin.last t) ≤ n) } (A n)) :
    (fun (n : ℕ) => (A n - n + n / Real.log n : ℝ)) =o[atTop] fun n => n / Real.log n := byA:ℕ → ℕhA:∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n ! = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n), t + 1 = x} (A n)⊢ (fun n => ↑(A n) - ↑n + ↑n / Real.log ↑n) =o[atTop] fun n => ↑n / Real.log ↑n
  sorryAll goals completed! 🐙

Cambie has observed that a positive answer follows from the result above with $a_t \leq n$, simply by pairing variables together, e.g. taking $a'i = a{2i-1}a_{2i}$ (and the lower bound follows from Stirling's approximation).

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_392.variants.implication (h : type_of% erdos_392) :
    type_of% erdos_392.variants.lower := byh:∀ (A : ℕ → ℕ),
  (∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n ! = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) →
    (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[atTop] fun n => ↑n / Real.log ↑n⊢ ∀ (A : ℕ → ℕ),
  (∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n ! = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n), t + 1 = x} (A n)) →
    (fun n => ↑(A n) - ↑n + ↑n / Real.log ↑n) =o[atTop] fun n => ↑n / Real.log ↑n
  sorryAll goals completed! 🐙


end Erdos392