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Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of new frozen rightmost digits returned by the integer tetration ${^{b}a}$ for any unitary increment of its hyperexponent $b \in \mathbb{Z}^{+}$ (for details and strict definitions, see https://arxiv.org/abs/2208.02622 and also https://nntdm.net/volume-28-2022/number-3/441-457/), assuming that $b$ is sufficiently large (i.e., a sufficient but not necessary condition for having a unique value of $V(a)$ - given $a$ - is $b \geq a+1$).

Now, let us call this fixed value of $V(a)$, which does not depend on the hyperexponent $b$, constant congruence speed of the tetration base $a$.

Then, from the aforementioned papers, we know that it is possible to find the exact value of the constant congruence speed of any tetration base $a \in \mathbb{Z}^{+} : a \not\equiv 0 \pmod{10}$ by simply calculating the $2$-adic or $5$-adic valuation of $a-1$ or $a+1$, or the $5$-adic order of $a^2+1$, depending on the congruence class of $a$ modulo $20$.

Thus, my question is pretty simple.
Which concrete applications or new ideas can be deduced from such an unexplored property of hyper-$4$ (i.e., the constancy of the congruence speed of the integer tetration ${^{b}a}$ and its map, having also proved the existence of infinitely many prime numbers characterized by any strictly positive integer value of the constant congruence speed)?

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    $\begingroup$ I find this too close to our Meta on how not to ask: “mathoverflow.net/help/dont-askl”. This could have multiple answers being equally valid; it lacks an actual problem to be solved; and the question is open-ended. $\endgroup$
    – user44143
    Commented Aug 8, 2022 at 2:03
  • $\begingroup$ I see... I posted this since usually we lose properties by climbing the ladder of hyperoperators, from multiplication (i.e., hyper-$2$) and above, but this is a very unique feature of hyper-4, that we cannot find in any higher-order operator, such as pentation. Now, my first thought has been some kind of cryptographic algorithm or a way to compute faster the congruences modulo $10^n$ of big numbers expressed as power towers, maybe. Just to give you a concrete application, we can know in a second how many digits of Graham's number are frozen, since it is a tetration base $3$ and $V(3)=1$. $\endgroup$
    – Marco Ripà
    Commented Aug 8, 2022 at 2:42

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