Primes in $\lfloor a^{n} \rfloor$

Solution 1:

This sequence not necessarily contains a prime number.

We will prove that there is a number $a\in [4,5]$ such that $\lfloor a^n \rfloor$ is even for all $n$.

The idea is that for $a\in[4,5]$, the set of possible values of $a^2$ is the interval $[4^2,5^2]$, which contains a even number. So, in the interval $[4,5]$ there is a subrange of values for $a$ such that $\lfloor a^2 \rfloor$ is even. Now, in that subrange, there will exist another subrange such that $\lfloor a^3 \rfloor$ is even for $a$ in this subrange, and so on. If we intersect all these subranges and use the nested intervals theorem, we will find a real $a$ such that all $\lfloor a\rfloor, \lfloor a^2\rfloor,\lfloor a^3\rfloor...$ are even.

Of course this is just the idea, see below the formal proof (I hope it is not too confusing):

Starting with the interval $I_1=[4.1,4.9]$ (we choose this initial interval to make sure $a$ is not an integer), we will construct inductively a infinite sequence of closed intervals $I_1,I_2,...$ satisfying the following conditions:

1) $ I_{n} \subset I_{n-1}$

2) If $I_n=[a_n,b_n]$, then $b_n^{n+1}-a_n^{n+1}\ge3$

3) The interval $[a_n^n,b_n^n]$ is contained in a interval of the form $[2k,2k+0.9]$ for some integer $k$.

The basis of induction is easily verified. Now, given $I_1,I_2,...I_n$, let's construct $I_{n+1}$:

Since $b_n^{n+1}-a_n^{n+1}\ge3$, the interval $[a_n^{n+1},b_n^{n+1}]$ must contain a interval of the form $[2k,2k+1]$ for some integer $k$. Then define $I_{n+1}$ to be $[a_{n+1},b_{n+1}]$ where $a_{n+1}=\sqrt[n+1]{2k}$ and $b_{n+1}=\sqrt[n+1]{2k+0.9}$.

3) Clearly holds for this interval

1) Holds because $a_{n+1}=\sqrt[n+1]{2k}\ge \sqrt[n+1]{a_n^{n+1}}=a_n$, and

$b_{n+1}=\sqrt[n+1]{2k+0.9}<\sqrt[n+1]{2k+1}\le \sqrt[n+1]{b_n^{n+1}}=b_n$

Finally, 2) holds because

$b_{n+1}^{n+2}-a_{n+1}^{n+2}= (2k+1)b_{n+1}-2ka_{n+1}=2k(b_{n+1}-a_{n+1})+b_{n+1}>b_{n+1}>4$ (because

$I_{n+1}\subset[4,5]$)

So we are done with our induction. Now, by the nested intervals theorem, there must exist a real number $a$ such that $a\in I_n$ for all $n$. But if $a\in I_n$, $then a^n\in [a_n^n,b_n^n]$. Since the last is contained in a interval of the form $[2k,2k+1]$, it follows that for this $a$, $\lfloor a^n \rfloor$ is even for all $n$.