12/03/2025, 04:13 AM
(This post was last modified: 01/28/2026, 11:05 AM by marcokrt.
Edit Reason: Adding more info and improving the content
)
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 \({v_b}^{[r]}(r^{r-1}-1)^r={v_b}^{[r]}(r^r-1)^r=r\).
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 \({v_b}^{[r]}(r^{r-1}-1)^r={v_b}^{[r]}(r^r-1)^r=r\).
Let \(G(n)\) be a generic reverse-concatenated sequence. If \(G(1) \notin \{2, 3, 7\}\), then \(^{G(n)}G(n) \pmod {10^d}≡^{G({n+1})}G({n+1}) \pmod {10^d}\), \(\forall n \in \mathbb{N}-\{0\}\)
("La strana coda della serie n^n^...^n", p. 60).
("La strana coda della serie n^n^...^n", p. 60).