Skip to content

Erdős problem 145

Start work

Let s1<s2<⋯s_1 < s_2 < \cdots be the sequence of squarefree numbers. Is it true that, for any α≥0\alpha\geq 0, lim⁡x→∞1x∑sn≤x(sn+1−sn)α \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha exists?

Sources

The source's own material, retained exactly.

4 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

145.lean

Retained formal statement4 of 4

Erdős [Er51] proved this for all 0≤α≤20\leq \alpha\leq 2.

[Er51] Erdös, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.

FormalConjectures/ErdosProblems/145.leanErdos145.erdos_145.variants.le_two5 linesExact file
∀ {α : ℝ},  α ∈ Set.Icc 0 2 →    ∃ β,      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop        (nhds β)
SolvedStatement only, no proof