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module
public import FormalConjecturesUtil
public import FormalConjectures.ErdosProblems.«246»Erdős Problem 1110
Reference: Erdős Problem 1110
@[expose] public sectionnamespace Erdos1110$n$ is representable with respect to $p$ and $q$ if it is the sum of a finite divisibility antichain of terms of the form $p^kq^l$.
def Representable (p q n : ℕ) : Prop :=
∃ s : Finset ℕ,
(s : Set ℕ) ⊆ Erdos246.Gamma p q ∧
IsAntichain (· ∣ ·) (s : Set ℕ) ∧
s.sum id = nLet $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.
If ${p,q}\neq {2,3}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers, that is, an infinite family of pairwise coprime non-representable integers?
@[category research open, AMS 5 11]
theorem erdos_1110 :
answer(sorry) ↔ ∀ (p q : ℕ), q < p → 2 ≤ q →
Nat.Coprime p q → ¬(p = 3 ∧ q = 2) →
∃ A : Set ℕ, A.Infinite ∧ A.Pairwise Nat.Coprime ∧
∀ n ∈ A, ¬Representable p q n := ⊢ True ↔
∀ (p q : ℕ),
q < p →
2 ≤ q → p.Coprime q → ¬(p = 3 ∧ q = 2) → ∃ A, A.Infinite ∧ A.Pairwise Nat.Coprime ∧ ∀ n ∈ A, ¬Representable p q n
All goals completed! 🐙end Erdos1110