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The congruence speed of tetration
#1
Hi everyone,

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?).

P.S. I have not defined the value of the congruence speed for the base 0, even if we could modify the definition in order to state that V(0)=0, since (for any positive integer n) 0^(2n)=1, while 0^(2n+1)=0 (using limits it would be ok)... thus, for any positive value of the hyperexponent, the number of new stable digits at the end of the result is 0. Would it be interesting?
Let G(n) be a generic reverse-concatenated sequence. If G(1)≠{2, 3, 7}, [G(n)^^G(n)](mod 10^d)≡[G(n+1)^^G(n+1)](mod 10^d), ∀n∈N\{0} (La strana coda della serie n^n^...^n, 60).
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#2
Looks very interesting!

I'll be sure to chop out some time to read these.

Regards, James
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#3
(12/17/2021, 01:09 AM)JmsNxn Wrote: Looks very interesting!

I'll be sure to chop out some time to read these.

Regards, James

Thank you very much, James.
I've just uploaded a basic preprint on RG where we use the congruence speed formula in order provide a complete answer to the question posted by Luknik here: Repetition of the last digits of a tetration of generic base

Here is the preprint we are working on, which I'd like to share just in case that you and/or somebody else are/is still interested in the topic: Number of stable digits of any integer tetration

Wishing all the forum members a happy new year (despite these difficult times)!

Marco
Let G(n) be a generic reverse-concatenated sequence. If G(1)≠{2, 3, 7}, [G(n)^^G(n)](mod 10^d)≡[G(n+1)^^G(n+1)](mod 10^d), ∀n∈N\{0} (La strana coda della serie n^n^...^n, 60).
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