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import FormalConjecturesUtilErdős Problem 647
namespace Erdos647
open Filter ArithmeticFunction.sigmaLet $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that $$ \max_{m < n}(m + \tau(m)) \leq n + 2? $$
@[category research open, AMS 11]
theorem erdos_647 : answer(sorry) ↔ ∃ n > 24, ⨆ m : Fin n, m + σ 0 m ≤ n + 2 := ⊢ True ↔ ∃ n > 24, ⨆ m, ↑m + (σ 0) ↑m ≤ n + 2
All goals completed! 🐙This is true for $n = 24$.
@[category research solved, AMS 11]
theorem erdos_647.variants.twenty_four : ⨆ m : Fin 24, (m : ℕ) + σ 0 m ≤ 26 := ⊢ ⨆ m, ↑m + (σ 0) ↑m ≤ 26
exact ciSup_le <| ⊢ ∀ (x : Fin 24), ↑x + (σ 0) ↑x ≤ 26 All goals completed! 🐙Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$ lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty. $$
@[category research open, AMS 11]
theorem erdos_647.variants.lim :
answer(sorry) ↔ atTop.Tendsto (fun n ↦ ⨆ m : Fin n, σ 0 m + m - n) atTop := ⊢ True ↔ Tendsto (fun n => ⨆ m, (σ 0) ↑m + ↑m - n) atTop atTop
All goals completed! 🐙Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which $$ \max_{n−k < m < n}(m + \tau(m)) ≤ n + 2. $$
@[category research open, AMS 11]
theorem erdos_647.variants.infinite :
answer(sorry) ↔ ∀ k, { n | ⨆ m : Set.Ioo (n - k) n, ↑m + σ 0 m ≤ n + 2 }.Infinite := ⊢ True ↔ ∀ (k : ℕ), {n | ⨆ m, ↑m + (σ 0) ↑m ≤ n + 2}.Infinite
All goals completed! 🐙
end Erdos647