Erdős problem 264 · problems.science

Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $\liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0$ is not an irrationality sequence.

Sources

Browse retained paths and inspect the exact material available for this Problem.

5 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

264.lean

Retained formal statement5 of 5

On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with $\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n \sim F(n)$ .

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

FormalConjectures/ErdosProblems/264.leanErdos264.erdos_264.variants.ko_tao_pos3 linesExact file

1∀ {F : ℕ → ℕ},2  Filter.Tendsto (fun n => ↑(F (n + 1)) / ↑(F n)) Filter.atTop Filter.atTop →3    ∃ a, Erdos264.IsIrrationalitySequence a ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(a n)) fun n => ↑(F n)

SolvedStatement only, no proofformal statement reference

Sources

Search problems.science