TITLE: Do the last digits of exponential towers really converge to a fixed sequence?
QUESTION [11 upvotes]: While fooling around with exponential towers I noticed something odd:
$$ 3^{3} \equiv 2\underline{7} \mod 100000  $$
$$ 3^{3^{3}} \equiv 849\underline{87} \mod 100000  $$
$$ 3^{3^{3^{3}}} \equiv 39\underline{387} \mod 100000  $$
$$ 3^{3^{3^{3^{3}}}} \equiv 5\underline{5387} \mod 100000  $$
$$ 3^{3^{3^{3^{3^{3}}}}} \equiv \underline{95387} \mod 100000  $$
$$ \dots$$
It seems that the last digits always converge to a fixed sequence! Is this really true and if yes - can someone think of a proof for this statement?
Any kind of help will be appreciated

REPLY [3 votes]: (Adding this answer to showcase a noteworthy generalization ...)
Yours is a very special case of the following:

If $k$ and $a_1,a_2,a_3,\ldots$ are any positive integers, then the sequence $$a_1,a_1^{a_2},a_1^{a_2^{a_3}},a_1^{a_2^{a_3^{a_4}}},\dots$$ is eventually constant $\pmod k.$

The above is a consequence of the following remarkable fact, which is proved as Corollary 2.11 in the paper Iterated exponents in number theory by D. B. Shapiro & S. D. Shapiro; Integers (2007), Vol. 7(1) #A23:

If $a_j\in\mathbb{Z_+}$, then the towers $a_1\uparrow a_2\uparrow\dots a_{h(k)}\uparrow  c$ reduce to the same value in $\mathbb{Z}/k\mathbb{Z}$, for every $c\gt 1.$ Consequently, $a_1\uparrow a_2\uparrow\dots a_{s}\uparrow x\pmod k$ is independent of the value of $x\in\mathbb{Z_+}$, provided $s\gt h(k)$.

Here $\ h(k) := \min\{s: \lambda^s(k) = 1\},$ where $\lambda()$ is the Carmichael function; e.g., $h(10^d)=d+2$ for $d\ge 1,$ as proved [here].
Thus, the corollary guarantees that at least $d$ rightmost decimal digits of $3\uparrow\uparrow n$ are "stable" if $n\gt d+2.$ (Your list suggests that $n\gt d+2$ is actually enough to stabilize $d+2$ digits.)