Checking this peculiar conjecture Are there infinitely many c-th perfect powers having a constant congruence speed of c?, a friend of mine has performed a computer search on the smallest tetration bases of the required form (i.e., \( 20 \cdot n + 5 \)) and the outcome shows that, as \(c \) grows, the rightmost digits of those smallest values \( 20 \cdot n + 5 \) freeze or \(10\)-adically converge to the string \(\dots 98869612812995910644531 \). So, the question now is to understand what decadic integer \(\dots 98869612812995910644531\) is, finding from which equation in the commutative ring of decadic integers it comes from.

Any hint?

Any hint?

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).