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import FormalConjecturesUtilErdős 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 {} := k:ℕ⊢ WeaklyDivisible k ∅
All goals completed! 🐙
A singleton is k-weakly divisible if k ≠ 0.
@[category API, AMS 11]
lemma weaklyDivisible_singleton {k : ℕ} (hk : k ≠ 0) (l : ℕ) : WeaklyDivisible k {l} := k:ℕhk:k ≠ 0l:ℕ⊢ WeaklyDivisible k {l}
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 := A:Finset ℕh:A.Nonempty⊢ ¬WeaklyDivisible 0 A
simpa [WeaklyDivisible] using ⟨{_}, A:Finset ℕh:A.Nonempty⊢ {?m.20} ⊆ A ∧ #{?m.20} = 1 ∧ (↑{?m.20}).Pairwise Nat.Coprime All 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 := ⊢ ∀ (k : ℕ), MaxWeaklyDivisible 0 k = 0
k:ℕ⊢ MaxWeaklyDivisible 0 k = 0
All goals completed! 🐙k:ℕhk:k ≠ 0this:{x | ∃ A, WeaklyDivisible k A ∧ (A = ∅ ∨ A = {1}) ∧ #A = x} = {0, 1}⊢ MaxWeaklyDivisible 1 k = 1
All goals completed! 🐙@[category test, AMS 11]
theorem maxWeaklyDivisible_zero_k (N : ℕ) : MaxWeaklyDivisible N 0 = 0 := N:ℕ⊢ MaxWeaklyDivisible N 0 = 0
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)k:ℕn:ℕh:n < khprime:Nat.nth Nat.Prime n = 1this:Nat.Prime 1⊢ False
All goals completed! 🐙@[category test, AMS 11]
theorem firstPrimesMultiples_zero_k_card_zero (N : ℕ) : (FirstPrimesMultiples N 0).card = 0 := N:ℕ⊢ #(FirstPrimesMultiples N 0) = 0
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.
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 + 1hcard_fink:Fintype.card (Fin k) = kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑yh_coprime:(↑x).Coprime ↑yp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime php_div_x:p ∣ ↑xhp_div_y:p ∣ ↑yhp_div_gcd:p ∣ 1h_gcd_eq_one:(↑x).gcd ↑y = 1hp_le_one:p ≤ 1⊢ False
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 + 1hcard_fink:Fintype.card (Fin k) = kx:↥sy:↥shne:x ≠ yheq:f x = f yh_pair:(↑s).Pairwise Nat.Coprimehne_val:↑x ≠ ↑yh_coprime:(↑x).Coprime ↑yp:ℕ := Nat.nth Nat.Prime ↑(f x)hp_prime:Nat.Prime php_div_x:p ∣ ↑xhp_div_y:p ∣ ↑yhp_div_gcd:p ∣ 1h_gcd_eq_one:(↑x).gcd ↑y = 1hp_le_one:p ≤ 1this:p > 1⊢ False
All 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 erdos_56 : answer(False) ↔ ∀ᵉ (k > 0) (N ≥ (k-1).nth Nat.Prime),
(MaxWeaklyDivisible N k = (FirstPrimesMultiples N k).card) := ⊢ False ↔ ∀ k > 0, ∀ N ≥ Nat.nth Nat.Prime (k - 1), MaxWeaklyDivisible N k = #(FirstPrimesMultiples N k)
All goals completed! 🐙end Erdos56