Erdős problem 299 · problems.science

Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?

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FormalConjectures/ErdosProblems/

299.lean

Retained formal statement1 of 2

Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?

There does not exist such a sequence, which follows from the positive solution to [erdosproblems.com/298] by Bloom [Bl21].

This was formalized in Lean 3 by Bloom and Mehta.

FormalConjectures/ErdosProblems/299.leanErdos299.erdos_2995 linesExact file

1False ↔2  ∃ a,3    StrictMono a ∧4      (∀ (n : ℕ), 0 < a n) ∧5        (fun n => ↑(a (n + 1)) - ↑(a n)) =O[Filter.atTop] 1 ∧ ∀ (S : Finset ℕ), ∑ i ∈ S, 1 / ↑(a i) ≠ 1

SolvedProof has a holeother systemexternal 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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