By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number.

Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?

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    $\begingroup$ I doubt it, but it's hard to prove anything. I think it's unknown whether, say, $[(8/7)^n]$ is composite infinitely often. $\endgroup$ – Gerry Myerson Dec 12 '15 at 12:09
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    $\begingroup$ @GerryMyerson Is it known that we get infinitely many composites for $[a^n]$ for some rational, non-integer $a$? I suspect no, but I'm quite curious. $\endgroup$ – Wojowu Dec 12 '15 at 21:23
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    $\begingroup$ @Woj, yes, it's known for $a=3/2$ and $a=4/3$, due to Forman and Shapiro, An arithmetic property of certain rational powers, Communications on Pure and Applied Mathematics Volume 20, Issue 3, pages 561–573, August 1967. More recently, there is work by Dubickas and by Novikas. See mathoverflow.net/questions/153426/… for some discussion. $\endgroup$ – Gerry Myerson Dec 12 '15 at 21:47

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