Repetition of the last digits of a tetration of generic base

[Luknik]

I found the the paper I wrote which extended the concept from tetration to the Ackermann function.
Repeating Digits

It's a very interesting result Daniel! Although my problem was to find the minimum hyper-exponent \(u \) of a natural base tetration such that the other tetrations with that base and with hyper-exponent greater or equal to \( u \) has the same last \( n \) digits. So I'm working to find a function \( f_{q}(n) \) such that if:
\(
f_{q}(n)=u
\)
Then for all \( m\geq u \)
\(
{^{m}q} \equiv {^{u}q} \mod (10^{n})
\)
where \(q \) is the base of the tetration and \(n \) is the number of the digits you want to be repeated.

Thank you for the avatar pic!