FormalConjectures.ErdosProblems.«421»

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import FormalConjecturesUtil

Erdős Problem 421

Reference: erdosproblems.com/421


open Set


namespace Erdos421

Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_421 : answer(sorry) ↔
    ∃ (d : ℕ → ℕ), StrictMono d ∧ 1 ≤ d 0 ∧ HasDensity (Set.range d) 1 ∧
    {(u, v) : ℕ × ℕ | u ≤ v}.InjOn fun (u, v) => ∏ i ∈ Finset.Icc u v, d i := by⊢ True ↔
  ∃ d,
    StrictMono d ∧
      1 ≤ d 0 ∧
        (range d).HasDensity 1 ∧
          InjOn
            (fun x =>
              match x with
              | (u, v) => ∏ i ∈ Finset.Icc u v, d i)
            {(u, v) | u ≤ v}
  sorryAll goals completed! 🐙


end Erdos421