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import FormalConjecturesUtilErdős Problem 394
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[ErHa78] Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485.
open Nat Filter Finsetopen scoped Asymptotics Topology Nat
namespace Erdos394
Let $t_k(n)$ denote the least $m$ such that $n\mid m(m+1)(m+2)\cdots (m+k-1).$
noncomputable def t (k n : ℕ) : ℕ :=
sInf { m : ℕ | 0 < m ∧ n ∣ ∏ i ∈ range k, (m + i) }
Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?
@[category research open, AMS 11]
theorem erdos_394.parts.i :
answer(sorry) ↔
∃ c > 0, (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
(t 2 n : ℝ)) ≪ (fun x ↦ x ^ 2 / (Real.log x) ^ c) := ⊢ True ↔ ∃ c > 0, (fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t 2 n)) =O[atTop] fun x => ↑x ^ 2 / Real.log ↑x ^ c
All goals completed! 🐙
Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$
@[category research open, AMS 11]
theorem erdos_394.parts.ii :
answer(sorry) ↔
∀ k ≥ 2, (fun (x : ℝ) ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
(t (k + 1) n : ℝ)) =o[atTop]
(fun (x : ℝ) ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
(t k n : ℝ)) := ⊢ True ↔ ∀ k ≥ 2, (fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t (k + 1) n)) =o[atTop] fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t k n)
All goals completed! 🐙
In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős and Hall [ErHa78], who proved that in fact $\sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2.$
@[category research solved, AMS 11]
theorem erdos_394.variants.hall_bound :
(fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊, (t 2 n : ℝ)) ≪
(fun x ↦ x ^ 2 * (Real.log (Real.log (Real.log x)) / Real.log (Real.log x))) := ⊢ (fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t 2 n)) =O[atTop] fun x =>
↑x ^ 2 * (Real.log (Real.log (Real.log ↑x)) / Real.log (Real.log ↑x))
All goals completed! 🐙
Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.
@[category research open, AMS 11]
theorem erdos_394.variants.hall_conjecture :
∀ c < Real.log 2, (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
(t 2 n : ℝ)) =o[atTop]
(fun x ↦ x ^ 2 / (Real.log x) ^ c) := ⊢ ∀ c < Real.log 2, (fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t 2 n)) =o[atTop] fun x => x ^ 2 / Real.log x ^ c
All goals completed! 🐙
Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}$.
@[category research solved, AMS 11]
theorem erdos_394.variants.lower_bound :
(fun x ↦ x ^ 2 / Real.log x) ≫
(fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊, (t 2 n : ℝ)) := ⊢ (fun x => ∑ n ∈ Icc 1 ⌊x⌋₊, ↑(t 2 n)) =O[atTop] fun x => ↑x ^ 2 / Real.log ↑x
All goals completed! 🐙
They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.
@[category research open, AMS 11]
theorem erdos_394.variants.factorial_gap_conjecture :
answer(sorry) ↔
Set.Infinite { n : ℕ | ∀ k, 2 ≤ k → k < n →
t k (n !) < t (k - 1) (n !) - 1 } := ⊢ True ↔ {n | ∀ (k : ℕ), 2 ≤ k → k < n → t k n ! < t (k - 1) n ! - 1}.Infinite
All goals completed! 🐙
They proved (with Selfridge) that this holds for $n=10$.
@[category research solved, AMS 11]
theorem erdos_394.variants.factorial_gap_10 :
∀ (k : ℕ), 2 ≤ k → k < 10 →
t k (10 !) <
t (k - 1) (10 !) - 1 := ⊢ ∀ (k : ℕ), 2 ≤ k → k < 10 → t k 10! < t (k - 1) 10! - 1
All goals completed! 🐙
end Erdos394