| import Mathlib.Data.Nat.Prime.Infinite |
| import Mathlib.Order.Interval.Finset.Nat |
| import OAI.NumberTheory.Ostmann.Defs |
|
|
| namespace OAI |
|
|
| /-! |
| # Sets in Ostmann's inverse Goldbach problem |
| |
| We use `ℕ` for the manuscript's nonnegative integers. The threshold convention |
| uses `N ≤ n`; changing finitely many initial values does not affect it. |
| -/ |
|
|
| namespace Ostmann |
|
|
| / |
| def EventuallyPrimeSumset (A B : Set ℕ) : Prop := |
| ∃ N : ℕ, ∀ n, N ≤ n → (n ∈ sumset A B ↔ n.Prime) |
|
|
| theorem sumset_comm (A B : Set ℕ) : sumset A B = sumset B A := by |
| ext n |
| simp only [mem_sumset] |
| constructor <;> rintro ⟨a, ha, b, hb, hab⟩ |
| · exact ⟨b, hb, a, ha, (Nat.add_comm b a).trans hab⟩ |
| · exact ⟨b, hb, a, ha, (Nat.add_comm b a).trans hab⟩ |
|
|
| theorem EventuallyPrimeSumset.symm {A B : Set ℕ} |
| (h : EventuallyPrimeSumset A B) : EventuallyPrimeSumset B A := by |
| simpa only [EventuallyPrimeSumset, sumset_comm B A] using h |
|
|
| open scoped symmDiff |
|
|
| / |
| theorem finite_symmDiff_iff (A B : Set ℕ) : |
| (sumset A B ∆ primes).Finite ↔ EventuallyPrimeSumset A B := by |
| constructor |
| · intro h |
| obtain ⟨M, hM⟩ := Set.finite_iff_bddAbove.mp h |
| refine ⟨M + 1, ?_⟩ |
| intro n hn |
| have hout : n ∉ sumset A B ∆ primes := by |
| intro hin |
| have := hM hin |
| omega |
| simp only [Set.mem_symmDiff, primes, Set.mem_ofPred_eq] at hout |
| tauto |
| · rintro ⟨N, hN⟩ |
| apply (Set.finite_Iio N).subset |
| intro n hn |
| by_contra hlt |
| have heq := hN n (Nat.le_of_not_gt hlt) |
| simp only [Set.mem_symmDiff, primes, Set.mem_ofPred_eq, heq, |
| and_not_self, or_self] at hn |
|
|
| / |
| theorem EventuallyPrimeSumset.prime_add {A B : Set ℕ} |
| (h : EventuallyPrimeSumset A B) : |
| ∃ N, ∀ a ∈ A, ∀ b ∈ B, N ≤ a + b → (a + b).Prime := by |
| obtain ⟨N, hN⟩ := h |
| exact ⟨N, fun a ha b hb hab => (hN (a + b) hab).mp ⟨a, ha, b, hb, rfl⟩⟩ |
|
|
| / |
| def TwoInfiniteSummandsImpossible : Prop := |
| ∀ A B : Set ℕ, A.Infinite → B.Infinite → ¬ EventuallyPrimeSumset A B |
|
|
| / |
| def InverseGoldbach : Prop := |
| ∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → (sumset A B ∆ primes).Infinite |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
