FormalConjectures.ErdosProblems.«3»

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import FormalConjecturesUtil

Erdős Problem 3

Reference: erdosproblems.com/3


namespace Erdos3

If $A \subset \mathbb{N} has $\sum_{n \in A}\frac 1 n = \infty$, then must $A$ contain arbitrarily long arithmetic progressions?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,
    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
    ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k := by⊢ True ↔ ∀ (A : Set ℕ), (¬Summable fun a => 1 / ↑↑a) → ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength ↑k
  sorryAll goals completed! 🐙


-- TODO(firsching): add the various known bounds as variants.

end Erdos3