Confirmed: the constancy of the congruence speed holds in each squarefree radix

[marcokrt]

It has been a long time since my last post here, so I hope you will forgive the sudden appearance after one year of silence.
I would like to share a very compact one-page note that summarizes six fundamental equations describing the constant congruence speed of integer tetration bases (greater than 1 and not a multiple of the selected radix) in arbitrary numeral systems \(r > 1\).
These few equations condense the key results established in our referenced recent works, including the complete formulas for prime radices, the explicit radix-6 and radix-10 cases, and a couple of elegant identities valid for all squarefree \(r > 2\).

In particular, I am especially fond of identity (4). It highlights the structural role of the \(r\)-adic fixed point \(1_r\): it considers 5 different parameters on the left-hand side, including the height of the power tower, which only needs to be greater than 1, and for \(k = 0\) (by Mihăilescu's theorem) it returns only perfect powers of exact order \(c\) whose radix-\(r\) constant congruence speed is exactly \(t\). Furthermore, it includes hyper-1 as sum/difference, hyper-2 as product, hyper-3 as exponentiation, and finally the function \(V\) arises from tetration (i.e., hyper-4).

I am attaching these formulas as a figure; the one-page note is also available on Zenodo: Six fundamental equations for the constant congruence speed in any numeral system.

Comments, questions, and suggestions are of course welcome... it may also be interesting to discuss possible extensions or implications for related r-adic fixed-point structures or iterated-exponentiation phenomena (I have also developed a very rough key-exchange idea based on solving fundamental equations of the form \(y^t = y\) for sufficiently large odd values of \(t := t\)(\(r\)) in the ring of \(r\)-adic integers \(\mathbb{Z}_r\)).

[marcokrt]

Let the integer \(r > 2\) indicate the considered base-\(r\) numeral system (as usual). Let \(a > 1\) and \(b > 1\) be two integers. Then, in radix-\(r\), the we call congruence speed of \(a\) at height \(b\) the number of the rightmost digits of \(^{b}a\) that do not change by moving to \(^{b+1}a\) and that weren't already stable digits at height \(b-1\). We denote by \({v_b}^{[r]}(a)\) the radix-\(r\) congruence speed of \(a\) at height \(b\).

More formally, for each integer \(b \geq 2\), let \(s_b\) be the largest integer such that \({}^{\,b+1}a \equiv {}^{\,b}a \pmod{r^{\,s_b}}\) and \({}^{\,b+1}a \not\equiv {}^{\,b}a \pmod{r^{\,s_b+1}}\): the difference \(s_b - s_{b-1}\) is the radix-\(r\) congruence speed of \(a\) at height \(b\).

Now, let \(q>0\) be an integer and denote by \(\nu_r(q)\) the maximum number of times that \(r\) divides \(q\) (i.e., the maximum integer \(m\) such that \(r^m\) divides \(q\)).
For every squarefree integer \(r > 2\) (and for all non-squarefree integers as well, except in some particular cases depending on both \(r\) and \(c\), together), for every integer \(q > 0\), for every integer \(k \geq 0\), for every integer \(b > 1\), and for every integer \(t > \nu_r(q) + 1\), the following identity hold:

\[{v_b}^{[r]}((k \cdot r^{t+1}+r^{t-\nu_r(q)}+1)^q)=t={v_b}^{[r]}((k \cdot r^{t+1}+r^{t-\nu_r(q)}-1)^q).\]
Here, \({v_b}^{[r]}((k \cdot r^{t+1}+r^{t-\nu_r(q)}+1)^q)=t\) comes from the solution \(1_r\) of the equation \(y^3=y\) in the ring of \(r\)-adic integers (for any given \(r>2\)), while \({v_b}^{[r]}((k \cdot r^{t+1}+r^{t-\nu_r(q)}-1)^q)=t\) comes from the specular solution of \(-1_r\) satisfying \(y^3=y\) in any given ring of \(r\)-adic integers such that \(r>2\).

In general, as long as \(b>1\), \(k \geq 0\), \(q>0\), and \(t > \nu_r(q)+1\),
\[t \leq {v_b}^{[r]}((k \cdot r^{t+1}+r^{t-\nu_r(q)} \pm 1)^q) \leq t+1\]
holds in every radix-\(r\) numeral system such that \(r>1\).
In general, \((k \cdot r^{t+1}+r^{t-\nu_r(q)} \pm 1)^q\) is characterized by a constant congruence speed as long as \(r>1\) is a squarefree integer (i.e., \(\mathrm{rad}\)(\(r\)) \(< r\), where \(\mathrm{rad}\)(\(r\)) denotes the largest squarefree integer dividing \(r\)).

Hence, assuming \(r > 2\) squarefree, \(b \geq 2\), and setting \(r=q=t\), we trivially get \({v_b}^{[r]}(r^{r-1}+1)^r={v_b}^{[r]}(r^r+1)=r\) and \({v_b}^{[r]}(r^{r-1}-1)^r={v_b}^{[r]}(r^r-1)^r=r\).