# Generalization of Mills' theorem

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?

• What does "smallest as possible" mean in the two bolded expressions? Jul 21, 2016 at 3:57
• You might find this paper of interest. Jul 21, 2016 at 4:01
• @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/… Jul 21, 2016 at 8:20

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: