Roland Waters

2022-06-20

Does there exist a positive irrational number $\alpha$, such that for any positive integer $n$ the number $⌊n\alpha ⌋$ is not a prime?

Abigail Palmer

This is called a Beatty sequence. There will indeed always be a prime in the sequence. The bound for the OP's sequence (provided $\alpha >1$) is
$p\le {L}^{35-16ϵ}{\alpha }^{2\left(1-ϵ\right)}{p}_{m+l}^{1+ϵ}$
where $L=\mathrm{log}\left(2\alpha \right)$, ${p}_{n}$ denotes the numerator of the ${n}^{\text{th}}$ convergent to the regular continued fraction expansion of $\alpha$, and $m$ is the unique integer such that ${p}_{m}\le {L}^{16}{\alpha }^{2}<{p}_{m+1}$. $ϵ$ can be chosen arbitrarily small, but l depends on $ϵ$.

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