Why is it not known if Mill's constant is rational or irrational?

The following text appears in the Mill’s constant definition at the Wikipedia:

There is no closed-form formula known for Mills’ constant, and it is not even known whether this number is rational (Finch 2003).

And then refers to Finch: Library of Congress Cataloging in Publication Data , Finch, Steven R., 1959- Mathematical constants / Steven R. Finch.
p. cm.- (Encyclopedia of mathematics and its applications; v. 94)

But the only reference in Finch’s text to Mill’s constant being rational or irrational is this excerpt (in the text Mill’s constant is named $C$):

It is not known whether C must necessarily be irrational.

Just that and only that. No references, no explanation if I did not miss nothing when reading it.

Making some simple calculations it seemed that it should be only irrational, but according to Finch that is wrong. This is my logic (the questions are at the end):

  1. Any Mill’s constant $A$ provides a prime-generating function $\lfloor A^{k^n} \rfloor$ for $k \gt 2$ where the constant is gradually calculated for each $n$ as $A_n=p^{\frac{1}{k^n}}$ for some selected on purpose prime $p \in [n^k,(n+1)^k]$ being the final unknown value of $A=lim_{n \to \infty}A_n$.

  2. If $A_n$ is rational then $A_n = p^{\frac{1}{k^n}} = \frac{a}{b}$ for $a,b \in \Bbb N$. Then:

$$b \cdot p^{\frac{1}{k^n}} = a$$

  1. If both sides are powered to ${k^n}$ then:

$$b^{(k^n)} \cdot p = a^{(k^n)}$$

  1. But that is only possible if $p=c^{(k^n)}$ for some natural number $c$. And that is a contradiction because $p$ is always a prime selected on purpose for each iteration when constructing the constant, so it can not have divisors apart from $1$ and $p$ by the very definition of prime numbers. So according to this if the logic is correct, $A_n$ could not be rational, they just could be irrational, $\forall n$ and $\forall k \gt 2$.

I would like to ask the following questions:

  1. Are the calculations correct? is the proof by contradiction wrong or it is not possible to apply it to $n \to \infty$?

  2. Why does Finch state that it is not known if $A$ must necessarily be irrational? Are there other references / papers regarding this point? Thank you!

Solutions Collecting From Web of "Why is it not known if Mill's constant is rational or irrational?"

(Answering 1. only:)

You have showed that the numbers $A_n$ cannot be rational. But Mills’ constant is the limit of that sequence of numbers, not one of the numbers themselves.

There is nothing to prevent a sequence of irrational numbers from having a rational limit. For a simpler example, consider the sequence $c_n=2^{1/n}$, with $\lim_{n \to \infty} c_n=1$.