I learned this interesting theorem but cannot remember from where since a long time has passed. What I'm wondering now is: Does someone know some constants which this theorem holds? ie, give some values of $a$, such that $[a^n]$ is **always prime** for $n\in N^+$(where $[\cdots]$ denotes integral part of a real number and $N^+$ is the set of positive integers).



Constant $a$ such that $[a^n]$ is always prime for $n\in N^+$
According to [Dubickas](http://www.mif.vu.lt/~dubickas/files/dvifai/monatshefte.pdf), the problem was still open in 2009; the conjecture is that no such $\alpha$ exists. Dubickas and his collaborators have worked extensively on integer parts of powers. See also [Baker&Harman](https://link.springer.com/article/10.1007%2FBF02622100) for a similar problem and more results in the negative direction.


