I was messing around with tetration sequences, i.e. sequences of the form,
\begin{equation} x_n = x \uparrow \uparrow n \mod{c} \end{equation}
where $x$, $c$, and $n$ are integer numbers. I noticed that these sequences seem to stabilize to a fixed value after some point. A few examples,
\begin{align}
2 &= 2 \mod{3}\\
2^2 &= 1 \mod{3}\\
2^{2^2} &= 1 \mod{3}\\
&\vdots \\
2\uparrow\uparrow n &= 1 \mod{3}, n \geq 2
\end{align}
\begin{align}
2 &= 2 \mod{5}\\
2^2 &= 4 \mod{5}\\
2^{2^2} &= 1 \mod{5}\\
&\vdots \\
2\uparrow\uparrow n &= 1 \mod{5}, n \geq 3
\end{align}
\begin{align}
2 &= 2 \mod{7}\\
2^2 &= 4 \mod{7}\\
2^{2^2} &= 2 \mod{7}\\
&\vdots \\
2\uparrow\uparrow n &= 2 \mod{7}, n \geq 3
\end{align}
The question : Do all sequences of this form eventually converge towards a fixed point (or maybe towards an alternating set of values ?
