What is a sufficient condition for the equivalence
$$ a_1 \uparrow a_2 \uparrow ... \uparrow a_n \equiv a_1 \uparrow a_2 \uparrow ... \uparrow a_n \uparrow a_{n+1}\ mod(\ m)\ ? $$
In a closely related question I got the answer that
$\phi^n(m) = 1$ is enough. Is this right in all cases ?
If k is the least number with $\phi^k(m) = 1$, I also know the bounds
$$\frac{log(m)}{log(3)}\le\ k\ \le\frac{log(m)}{log(2)}$$
Can a condition be derived with the help of these bounds ?
I also tried induction but the induction step is difficult because $a^m \equiv a^n$ (mod k) only implies $m \equiv n$ mod ($\ \phi(k)$), if $gcd(a,k)=1$. I could not handle the general case.