FormalConjecturesForMathlib.NumberTheory.Divisors

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module

public import Mathlib.Data.Nat.Nth
public import Mathlib.NumberTheory.Divisors

@[expose] public section

The increasing enumeration of the divisors of a natural number

Basic facts about Nat.nth (· ∈ n.divisors), the increasing enumeration of the divisors of n.

namespace Nat

The smallest divisor of a positive number is 1, i.e. the 0th entry of the increasing enumeration of its divisors.


lemma nth_divisors_zero {n : ℕ} (hn : n ≠ 0) : Nat.nth (· ∈ n.divisors) 0 = 1 := byn:ℕhn:n ≠ 0⊢ nth (fun x ↦ x ∈ n.divisors) 0 = 1
  rw [Nat.nth_zeron:ℕhn:n ≠ 0⊢ sInf {x | x ∈ n.divisors} = 1n:ℕhn:n ≠ 0⊢ sInf {x | x ∈ n.divisors} = 1]n:ℕhn:n ≠ 0⊢ sInf {x | x ∈ n.divisors} = 1
  exact IsLeast.csInf_eq ⟨Nat.one_mem_divisors.mpr hn, fun y hy => Nat.pos_of_mem_divisors hy⟩All goals completed! 🐙

Every divisor enumerated after index 0 is at least 2.


lemma two_le_nth_divisors {n : ℕ} (hn : n ≠ 0) {i : ℕ} (hi : i ≠ 0)
    (h : Nat.nth (· ∈ n.divisors) i ≠ 0) : 2 ≤ Nat.nth (· ∈ n.divisors) i := byn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i
  have hfin : (Set.ofPred (· ∈ n.divisors)).Finite := n.divisors.finite_toSetn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finite⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i
  have hpos : 1 ≤ Nat.nth (· ∈ n.divisors) i := Nat.pos_of_mem_divisors (Nat.nth_mem_of_ne_zero h)n:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) i⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i
  rcases hpos.lt_or_eq with h2 | h1inln:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih2:1 < nth (fun x ↦ x ∈ n.divisors) i⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) iinrn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i
  ·inln:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih2:1 < nth (fun x ↦ x ∈ n.divisors) i⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i omegaAll goals completed! 🐙
  ·inrn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i exact absurd (Nat.nth_injOn hfin
      (Set.mem_Iio.mpr (Nat.lt_card_toFinset_of_nth_ne_zero h hfin))
      (Set.mem_Iio.mpr (Nat.lt_card_toFinset_of_nth_ne_zero
        (show Nat.nth (· ∈ n.divisors) 0 ≠ 0 byn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0⊢ 2 ≤ nth (fun x ↦ x ∈ n.divisors) i rw [nth_divisors_zero hnn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ 1 ≠ 0n:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ 1 ≠ 0]n:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ 1 ≠ 0; omegaAll goals completed! 🐙) hfin))
      (byn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ nth (fun x ↦ x ∈ n.divisors) i = nth (fun x ↦ x ∈ n.divisors) 0 rw [nth_divisors_zero hnn:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ nth (fun x ↦ x ∈ n.divisors) i = 1n:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ nth (fun x ↦ x ∈ n.divisors) i = 1]n:ℕhn:n ≠ 0i:ℕhi:i ≠ 0h:nth (fun x ↦ x ∈ n.divisors) i ≠ 0hfin:{x | x ∈ n.divisors}.Finitehpos:1 ≤ nth (fun x ↦ x ∈ n.divisors) ih1:1 = nth (fun x ↦ x ∈ n.divisors) i⊢ nth (fun x ↦ x ∈ n.divisors) i = 1; omegaAll goals completed! 🐙)) hi

end Nat