FormalConjectures.ErdosProblems.«351»

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import FormalConjecturesUtil

Erdős Problem 351

Reference: erdosproblems.com/351

open Polynomialnamespace Erdos351

The set of rational numbers of the form P(n) + 1 / n where n is a natural number and P is a polynomial with rational coefficients.

Note: We include P 0 in there (since 1 / 0 = 0), but this doesn't change the validity of the conjecture

def imageSet {α : Type*} [Semifield α] (P : α[X]) : Set α :=
  Set.range (fun (n : ℕ) ↦ P.eval ↑n + 1 / n)

The predicate that a set A is strongly complete, i.e. that for every finite set B, every sufficiently large integer is a sum of elements of the set A \ B.

def IsStronglyComplete {α : Type*} [Semiring α] (A : Set α) : Prop :=
  ∀ B : Finset α,
    ∀ᶠ (m : ℕ) in Filter.atTop,
      ↑m ∈ { ∑ n ∈ X, n | (X : Finset α) (_ : ↑X ⊆ A \ B) }

The predicate that the rational polynomial P has a complete image.

def HasCompleteImage (P : ℚ[X]) : Prop := IsStronglyComplete (imageSet P)

Let $p(x) \in \mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading coefficient. Is it true that $$A={ p(n)+1/n : n \in \mathbb{N}}$$ is strongly complete, in the sense that, for any finite set $B$, $$\left{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right}$$ contains all sufficiently large integers?

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos351.lean"]
theorem erdos_351 :
    answer(True) ↔ ∀ P : ℚ[X], 0 < P.natDegree → 0 < P.leadingCoeff → HasCompleteImage P := by⊢ True ↔ ∀ (P : ℚ[X]), 0 < P.natDegree → 0 < P.leadingCoeff → HasCompleteImage P
  sorryAll goals completed! 🐙

Let $p(x) = x \in \mathbb{Q}[x]$. It has been shown that $$A={ p(n)+1/n : n \in \mathbb{N}}$$ is strongly complete, in the sense that, for any finite set $B$, $$\left{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right}$$ contains all sufficiently large integers.

@[category research solved, AMS 11]
protected theorem erdos_351.variants.X : HasCompleteImage X := by⊢ HasCompleteImage X
  sorryAll goals completed! 🐙

Let $p(x) = x ^ 2 \in \mathbb{Q}[x]$. It has been shown that $$A={ p(n)+1/n : n \in \mathbb{N}}$$ is strongly complete, in the sense that, for any finite set $B$, $$\left{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right}$$ contains all sufficiently large integers.

@[category research solved, AMS 11]
theorem erdos_351.variants.X_sq : HasCompleteImage (X ^ 2) := by⊢ HasCompleteImage (X ^ 2)
  sorryAll goals completed! 🐙

end Erdos351