Examples of other Mills' constants

Question

Mills' constant is the well-known constant A such that the function $\lfloor A^{{3}^{n}} \rfloor$ gives primes for all natural numbers $n$, whose value is ~$1.306377883...$

It is also well-known that there is an infinity of functions $\lfloor A^{{r}^{n}} \rfloor$ that gives primes for all $n$, depending on the value of $r$.

Are there any examples of such $\lfloor A^{{r}^{n}} \rfloor$ where $r \neq 3$? There seems to be very little information available on this subject.

Answer 1

Given a particular value of $r \ge 3$, a high precision calculator and a method of finding the next prime above a number, it is easy to find the initial digits of $A_r$.

For example with $r=4$:

• start with the smallest prime $2$
• $2^4=16$
• the smallest prime above $16$ is $17$
• $17^4 = 83521$
• the smallest prime above $83521$ is $83537$
• etc.

We now want $2 \le A_4^{4^1} \lt 3$ and $17 \le A_4^{4^2} \lt 18$ and $83537 \le A_4^{4^3} \lt 83538$ and so on

This will give $A_4=1.193725\ldots$

Meanwhile, for $r=2$, then if Legendre's conjecture that there is a prime between $n^2$ and $(n+1)^2$ is true, you have $\lfloor 1.5246999605380943599233635756884211622202236231…^{2^n}\rfloor$ giving primes: see OEIS A059784