FormalConjectures.ErdosProblems.«56»

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import FormalConjecturesUtil

Erdős Problem 56

Reference: erdosproblems.com/56

open scoped Finsetnamespace Erdos56

Say a set of natural numbers is k-weakly divisible if any k+1 elements of A are not relatively prime.

def WeaklyDivisible (k : ℕ) (A : Finset ℕ) : Prop :=
    ∀ s ∈ A.powersetCard (k + 1), ¬ Set.Pairwise s Nat.Coprime


@[category API, AMS 11]
lemma weaklyDivisible_empty (k : ℕ): WeaklyDivisible k {} := byk:ℕ⊢ WeaklyDivisible k ∅
  simp [WeaklyDivisible]All goals completed! 🐙

A singleton is k-weakly divisble if k ≠ 0.

@[category API, AMS 11]
lemma weaklyDivisible_singleton {k : ℕ} (hk : k ≠ 0) (l : ℕ) : WeaklyDivisible k {l} := byk:ℕhk:k ≠ 0l:ℕ⊢ WeaklyDivisible k {l}
  simp [WeaklyDivisible, hk]All goals completed! 🐙

No non-empty set is 1-weakly divisible.

@[category API, AMS 11]
lemma not_weaklyDivisible_zero {A : _} (h : A.Nonempty) : ¬WeaklyDivisible 0 A := byA:Finset ℕh:A.Nonempty⊢ ¬WeaklyDivisible 0 A
  simpa [WeaklyDivisible] using ⟨{_}, byA:Finset ℕh:A.Nonempty⊢ {?m.20} ⊆ A ∧ #{?m.20} = 1 ∧ (↑{?m.20}).Pairwise Nat.Coprime simpa using h.choose_specAll goals completed! 🐙⟩


@[category API, AMS 11]
lemma empty_iff_weaklyDivisible_zero {A : _} : WeaklyDivisible 0 A ↔ A = ∅ :=
  ⟨fun h ↦ Finset.not_nonempty_iff_eq_empty.1 <| mt not_weaklyDivisible_zero (not_not.2 h),
    fun h ↦ h ▸ weaklyDivisible_empty _⟩

MaxWeaklyDivisible N k is the size of the largest k-weakly divisible subset of {1,..., N}

noncomputable def MaxWeaklyDivisible (N : ℕ) (k : ℕ) : ℕ :=
  sSup {#A | (A : Finset ℕ) (_ : A ⊆ Finset.Icc 1 N) (_ : WeaklyDivisible k A)}


@[category test, AMS 11]
theorem maxWeaklyDivisible_zero : ∀ k : ℕ, MaxWeaklyDivisible 0 k = 0 := by⊢ ∀ (k : ℕ), MaxWeaklyDivisible 0 k = 0
  intro kk:ℕ⊢ MaxWeaklyDivisible 0 k = 0
  simp [MaxWeaklyDivisible, Nat.sSup_def]All goals completed! 🐙


@[category test, AMS 11]
theorem maxWeaklyDivisible_one {k : ℕ} (hk : k ≠ 0) : MaxWeaklyDivisible 1 k = 1 := byk:ℕhk:k ≠ 0⊢ MaxWeaklyDivisible 1 k = 1
  have : {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x} = {0, 1} := byk:ℕhk:k ≠ 0⊢ MaxWeaklyDivisible 1 k = 1
    refine Set.ext fun _ => ⟨fun _ => byk:ℕhk:k ≠ 0x✝¹:ℕx✝:x✝¹ ∈ {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x}⊢ x✝¹ ∈ {0, 1} aesopAll goals completed! 🐙, ?_⟩
    rintro ⟨_, _⟩inlk:ℕhk:k ≠ 0⊢ 0 ∈ {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x}inrk:ℕhk:k ≠ 0x✝:ℕh✝:x✝ ∈ {1}⊢ x✝¹ ∈ {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x}
    ·inlk:ℕhk:k ≠ 0⊢ 0 ∈ {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x} simpa using weaklyDivisible_empty kAll goals completed! 🐙
    ·inrk:ℕhk:k ≠ 0x✝:ℕh✝:x✝ ∈ {1}⊢ x✝¹ ∈ {x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x} exact ⟨{1}, byk:ℕhk:k ≠ 0x✝:ℕh✝:x✝ ∈ {1}⊢ WeaklyDivisible k {1} ∧ ({1} = ∅ ∨ {1} = {1}) ∧ #{1} = x✝ simp_all [weaklyDivisible_singleton hk 1]All goals completed! 🐙⟩
  simp_all [MaxWeaklyDivisible]All goals completed! 🐙


@[category test, AMS 11]
theorem maxWeaklyDivisible_zero_k (N : ℕ) : MaxWeaklyDivisible N 0 = 0 := byN:ℕ⊢ MaxWeaklyDivisible N 0 = 0
  simp [empty_iff_weaklyDivisible_zero, MaxWeaklyDivisible]All goals completed! 🐙

FirstPrimesMultiples N k is the set of numbers in {1,..., N} that are a multiple of one of the first k primes.

noncomputable def FirstPrimesMultiples (N k : ℕ) : Finset ℕ :=
    (Finset.Icc 1 N).filter fun i => ∃ j < k, (j.nth Nat.Prime ∣ i)


@[category test, AMS 11]
theorem firstPrimesMultiples_one_card_zero (k : ℕ) : (FirstPrimesMultiples 1 k).card = 0 := byk:ℕ⊢ #(FirstPrimesMultiples 1 k) = 0
  simp [FirstPrimesMultiples, Finset.filter_singleton]k:ℕ⊢ ∀ x < k, ¬Nat.nth Nat.Prime x = 1
  intro n hk:ℕn:ℕh:n < k⊢ ¬Nat.nth Nat.Prime n = 1
  by_contra hprimek:ℕn:ℕh:n < khprime:Nat.nth Nat.Prime n = 1⊢ False
  have : Nat.Prime 1 := byk:ℕ⊢ #(FirstPrimesMultiples 1 k) = 0
    convert Nat.prime_nth_prime nh.e'_1k:ℕn:ℕh:n < khprime:Nat.nth Nat.Prime n = 1⊢ 1 = Nat.nth Nat.Prime n
    exact hprime.symmAll goals completed! 🐙
  tautoAll goals completed! 🐙


@[category test, AMS 11]
theorem firstPrimesMultiples_zero_k_card_zero (N : ℕ) : (FirstPrimesMultiples N 0).card = 0 := byN:ℕ⊢ #(FirstPrimesMultiples N 0) = 0
  simp [FirstPrimesMultiples]All goals completed! 🐙

An example of a k-weakly divisible set is the subset of {1, ..., N} containing the multiples of the first k primes.

@[category API, AMS 11]
lemma weaklyDivisible_firstPrimesMultiples (N k : ℕ) :
    WeaklyDivisible k (FirstPrimesMultiples N k) := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k)
  unfold WeaklyDivisibleN:ℕk:ℕ⊢ ∀ s ∈ Finset.powersetCard (k + 1) (FirstPrimesMultiples N k), ¬(↑s).Pairwise Nat.Coprime
  intro s hsN:ℕk:ℕs:Finset ℕhs:s ∈ Finset.powersetCard (k + 1) (FirstPrimesMultiples N k)⊢ ¬(↑s).Pairwise Nat.Coprime
  rw [Finset.mem_powersetCard] at hsN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1⊢ ¬(↑s).Pairwise Nat.Coprime
  -- Define the map from s to Fin k
  set f : s → Fin k := fun ⟨x, hx⟩ =>
    have h_exists := (Finset.mem_filter.mp (hs.1 hx)).2
    ⟨Classical.choose h_exists, (Classical.choose_spec h_exists).1⟩N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩⊢ ¬(↑s).Pairwise Nat.Coprime
  have hcard_s : Fintype.card s = k + 1 := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k) simp [hs.2]All goals completed! 🐙
  have hcard_fink : Fintype.card (Fin k) = k := Fintype.card_fin kN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin k⊢ ¬(↑s).Pairwise Nat.Coprime
  obtain ⟨x, y, hne, heq⟩ := Fintype.exists_ne_map_eq_of_card_lt f (byN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin k⊢ Fintype.card (Fin k) < Fintype.card ↥s omegaAll goals completed! 🐙)
  intro h_pairN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprime⊢ False
  have hne_val : x.1 ≠ y.1 := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k)
    intro h_eqN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimeh_eq:↑x = ↑y⊢ False
    exact hne (Subtype.ext h_eq)All goals completed! 🐙
  have h_coprime := h_pair x.2 y.2 hne_valN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_val⊢ False
  set p := (f x).val.nth Nat.PrimeN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)⊢ False
  have hp_prime : p.Prime := Nat.prime_nth_prime _N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)⊢ False
  have hp_div_x : p ∣ x.1 := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k)
    have h_exists := (Finset.mem_filter.mp (hs.1 x.2)).2N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)h_exists:∃ j < k, Nat.nth Nat.Prime j ∣ ↑x := (Finset.mem_filter.mp (hs.left x.property)).right⊢ p ∣ ↑x
    exact (Classical.choose_spec h_exists).2All goals completed! 🐙
  have hp_div_y : p ∣ y.1 := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k)
    have h_exists := (Finset.mem_filter.mp (hs.1 y.2)).2N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righth_exists:∃ j < k, Nat.nth Nat.Prime j ∣ ↑y := (Finset.mem_filter.mp (hs.left y.property)).right⊢ p ∣ ↑y
    have h_eq_f : (f x).val = (f y).val := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k) rw [heq]All goals completed! 🐙
    -- (f y).val is Classical.choose h_exists by definition
    have : p = (Classical.choose h_exists).nth Nat.Prime := byN:ℕk:ℕ⊢ WeaklyDivisible k (FirstPrimesMultiples N k)
      dsimp [p]N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righth_exists:∃ j < k, Nat.nth Nat.Prime j ∣ ↑y := (Finset.mem_filter.mp (hs.left y.property)).righth_eq_f:↑(f x) = ↑(f y) := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y))⊢ Nat.nth Nat.Prime ↑(f x) = Nat.nth Nat.Prime (Classical.choose h_exists)
      rw [h_eq_f]All goals completed! 🐙
    rw [this]N:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righth_exists:∃ j < k, Nat.nth Nat.Prime j ∣ ↑y := (Finset.mem_filter.mp (hs.left y.property)).righth_eq_f:↑(f x) = ↑(f y) := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y))this:p = Nat.nth Nat.Prime (Classical.choose h_exists) := 
  id
    (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
      (Eq.refl (Nat.nth Nat.Prime ↑(f y))))⊢ Nat.nth Nat.Prime (Classical.choose h_exists) ∣ ↑y
    exact (Classical.choose_spec h_exists).2All goals completed! 🐙
  have hp_div_gcd : p ∣ x.1.gcd y.1 := Nat.dvd_gcd hp_div_x hp_div_yN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righthp_div_y:p ∣ ↑y := 
  have h_exists := (Finset.mem_filter.mp (hs.left y.property)).right;
  have h_eq_f := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y));
  have this :=
    id
      (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
        (Eq.refl (Nat.nth Nat.Prime ↑(f y))));
  Eq.mpr (id (congrArg (fun _a => _a ∣ ↑y) this)) (Classical.choose_spec h_exists).righthp_div_gcd:p ∣ (↑x).gcd ↑y := Nat.dvd_gcd hp_div_x hp_div_y⊢ False
  have h_gcd_eq_one : x.1.gcd y.1 = 1 := h_coprimeN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righthp_div_y:p ∣ ↑y := 
  have h_exists := (Finset.mem_filter.mp (hs.left y.property)).right;
  have h_eq_f := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y));
  have this :=
    id
      (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
        (Eq.refl (Nat.nth Nat.Prime ↑(f y))));
  Eq.mpr (id (congrArg (fun _a => _a ∣ ↑y) this)) (Classical.choose_spec h_exists).righthp_div_gcd:p ∣ (↑x).gcd ↑y := Nat.dvd_gcd hp_div_x hp_div_yh_gcd_eq_one:(↑x).gcd ↑y = 1 := h_coprime⊢ False
  rw [h_gcd_eq_one] at hp_div_gcdN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righthp_div_y:p ∣ ↑y := 
  have h_exists := (Finset.mem_filter.mp (hs.left y.property)).right;
  have h_eq_f := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y));
  have this :=
    id
      (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
        (Eq.refl (Nat.nth Nat.Prime ↑(f y))));
  Eq.mpr (id (congrArg (fun _a => _a ∣ ↑y) this)) (Classical.choose_spec h_exists).righthp_div_gcd:p ∣ 1h_gcd_eq_one:(↑x).gcd ↑y = 1⊢ False
  have hp_le_one : p ≤ 1 := Nat.le_of_dvd (byN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righthp_div_y:p ∣ ↑y := 
  have h_exists := (Finset.mem_filter.mp (hs.left y.property)).right;
  have h_eq_f := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y));
  have this :=
    id
      (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
        (Eq.refl (Nat.nth Nat.Prime ↑(f y))));
  Eq.mpr (id (congrArg (fun _a => _a ∣ ↑y) this)) (Classical.choose_spec h_exists).righthp_div_gcd:p ∣ 1h_gcd_eq_one:(↑x).gcd ↑y = 1⊢ 0 < 1 norm_numAll goals completed! 🐙) hp_div_gcd
  have : p > 1 := hp_prime.one_ltN:ℕk:ℕs:Finset ℕhs:s ⊆ FirstPrimesMultiples N k ∧ #s = k + 1f:↥s → Fin k := 
  fun x =>
    match x with
    | ⟨x, hx⟩ =>
      have h_exists := ⋯;
      ⟨Classical.choose h_exists, ⋯⟩hcard_s:Fintype.card ↥s = k + 1 := 
  of_eq_true
    (Eq.trans
      (Eq.trans (congrArg (fun x => x = k + 1) (Eq.trans (Fintype.card_coe s) hs.right)) Nat.add_left_cancel_iff._simp_1)
      (eq_self 1))hcard_fink:Fintype.card (Fin k) = k := Fintype.card_fin kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑y := fun h_eq => hne (Subtype.ext h_eq)h_coprime:(↑x).Coprime ↑y := h_pair x.property y.property hne_valp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime p := Nat.prime_nth_prime ↑(f x)hp_div_x:p ∣ ↑x := 
  have h_exists := (Finset.mem_filter.mp (hs.left x.property)).right;
  (Classical.choose_spec h_exists).righthp_div_y:p ∣ ↑y := 
  have h_exists := (Finset.mem_filter.mp (hs.left y.property)).right;
  have h_eq_f := Eq.mpr (id (congrArg (fun _a => ↑_a = ↑(f y)) heq)) (Eq.refl ↑(f y));
  have this :=
    id
      (Eq.mpr (id (congrArg (fun _a => Nat.nth Nat.Prime _a = Nat.nth Nat.Prime (Classical.choose h_exists)) h_eq_f))
        (Eq.refl (Nat.nth Nat.Prime ↑(f y))));
  Eq.mpr (id (congrArg (fun _a => _a ∣ ↑y) this)) (Classical.choose_spec h_exists).righthp_div_gcd:p ∣ 1h_gcd_eq_one:(↑x).gcd ↑y = 1hp_le_one:p ≤ 1 := 
  Nat.le_of_dvd
    (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0))
      (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl false))
    hp_div_gcdthis:p > 1 := Nat.Prime.one_lt hp_prime⊢ False
  omegaAll goals completed! 🐙

Suppose $A \subseteq {1,\dots,N}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set? To avoid trivial counterexamples, we must insist that $N$ be at least the $k$th prime.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos56.lean"]
theorem declaration uses 'sorry'erdos_56 : answer(False) ↔ ∀ᵉ (k > 0) (N ≥ (k-1).nth Nat.Prime),
    (MaxWeaklyDivisible N k = (FirstPrimesMultiples N k).card) := by⊢ False ↔ ∀ k > 0, ∀ N ≥ Nat.nth Nat.Prime (k - 1), MaxWeaklyDivisible N k = #(FirstPrimesMultiples N k)
  sorryAll goals completed! 🐙


end Erdos56