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Mills' theorem states that there exists a positive real number $A$ such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers $n$. The value of $A$ is approximately $1.306$...., and primes generated by this constant A is $2,11,1361,....$, these are called as Mills' primes.

Now I want make some kind of generalization of this Mill's theorem: There exist 2 positive real numbers $B$ and $C$, such that the floor of the double exponential function $B^{C^n}$ are primes for all positive integers $n$. The values of $B$ and $C$ are chosen to be as smallest as possible, so it can generates the sequence of distinct increasing primes that are smallest as possible. I made an experiment and I have a problem of determining the values of $B$ and $C$, because it grows very fast. Does anyone able to determine the value of $B$ and $C$? What is the values of $B$ and $C$ might be?

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  • $\begingroup$ What does "smallest as possible" mean in the two bolded expressions? $\endgroup$ Commented Jul 21, 2016 at 3:57
  • $\begingroup$ You might find this paper of interest. $\endgroup$ Commented Jul 21, 2016 at 4:01
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    $\begingroup$ @senpuretsuzan you should review the original Mills paper to understand the values that can be used for $B$ and $C$. I wrote a similar question to yours here some weeks ago. I was able to reduce the growth of the sequence of primes with a little trick based on Mills results. math.stackexchange.com/questions/1807823/… $\endgroup$ Commented Jul 21, 2016 at 8:20

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If you want to find such positive real numbers $B$ and $C$, you first need to understand Mills' proof of his original prime-representing formula. It's only one page, you can read it freely online here:

PDF link to Mills' paper

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