What is this number (\(\dots 98869612812995910644531\))?

[marcokrt]

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?

[marcokrt]

Ok, problem solved. We just needed to resume the original tetration base by multiplying those numbers by \(20\), adding \(5\) to the result so that we find the corresponding solution from \(y^5=y\), which is the \(10\)-adic integer \( \alpha_{25}=\{5^{2^n}\}_{\infty}=\dots 92256259918212890625 \).