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).
- $5^n+n$ is never prime?
- Sum of odd prime and odd semiprime as sum of two odd primes?
- what is the name of this number? is it transcendental?
- There always exists a sequence of consecutive composite integers of length $n$ for all $n$.
- About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
- Prime Numbers and a Two-Player Game
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):
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 $A=lim_{n \to \infty}A_n$.
If $A_n$ is rational then $A_n = p^{\frac{1}{k^n}} = \frac{a}{b}$ for $a,b \in \Bbb N$. Then:
$$b \cdot p^{\frac{1}{k^n}} = a$$
$$b^{(k^n)} \cdot p = a^{(k^n)}$$
I would like to ask the following questions:
Are the calculations correct? is the proof by contradiction wrong or it is not possible to apply it to $n \to \infty$?
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!
(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 $\lim_{n \to \infty} c_n=1$.