Erdős problem 1102 · problems.science

There exists an infinite sequence $A = {a₁ < a₂ < …} ⊂ \mathsf{SF}$ where $\mathsf{SF} := \mathbb{N} \setminus \bigcup_{p} p^{2}\mathbb{N}$ , i.e. the set of squarefree numbers. The set A has property Q and natural density 6 / π^2. Equivalently, (j / a_j) → 6/π^2 as j → ∞. -

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

1102.lean

Retained formal statement2 of 4

Conversely, for any function f : ℕ → ℕ that goes to infinity, there exists a strictly increasing sequence A = {a₁ < a₂ < …} with property P such that (a_j / j) ≤ f(j) for all j. -

FormalConjectures/ErdosProblems/1102.leanErdos1102.erdos_1102.exists_sequence_with_P3 linesExact file

1∀ (f : ℕ → ℕ),2  Filter.Tendsto f Filter.atTop Filter.atTop →3    (∀ (n : ℕ), f n ≠ 0) → ∃ A, StrictMono A ∧ Erdos1102.HasPropertyP (Set.range A) ∧ ∀ (j : ℕ), ↑(A j) / ↑j ≤ ↑(f j)

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

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