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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2 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

299.lean

Retained formal statement2 of 2

The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21].

The statement is as follows: If $A \subset \mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive density) then there is a finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$ .

FormalConjectures/ErdosProblems/299.leanErdos299.erdos_299.variants.density1 lineExact file

1∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1

SolvedStatement only, no proof

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