I wish to share a couple of papers of mine that have recently been published, since they introduce a peculiar property which I named as the constancy of the "congruence speed" of (integer) tetration; it holds for any base that is not congruent to 0(mod 10).

In the very recent paper "The congruence speed formula" the whole map of the related function N\{0}-->NU{0} has been shown, and this is quite surprising due to the fascinating recurring patterns linked to the 15 solutions of the fundamental equation y^5=y in the commutative ring of 10-adic integers.

BTW, the last of the two papers, published just one week ago, can be read here (NNTDM, 27(4), pp. 43-61): The congruence speed formula, while the oldest one (NNTDM, 26(3), pp. 245-260) is here: On the constant congruence speed of tetration

Hoping that these original results can help us to know a little more about the beautiful caos induced by hyper-4, inspiring some original research papers by others in the near future (generalizing my formula outside radix-10, maybe?).