A Compact Notation for Peculiar Properties Characterizing Integer Tetration

By initially working in the decimal numeral system, we introduce a compact notation to express the congruence speed of an integer tetration base \(a\), along with the cycle of the rightmost non-stable digits of \(^{b}a\) for unit increments of \(b\). The resulting discrete function provides a useful tool for efficiently computing the exact number of frozen digits that characterize the right tail of each nontrivial integer tetration. We also establish an improved upper bound for the minimum hyperexponent \(\bar{b}(a)\) that guarantees the constancy of the congruence speed of \(a\) for all heights \(b \geq \bar{b}(a)\). Moreover, we prove that the minimum between the constant congruence speeds of any two integers greater than \(1\), whose product is not divisible by \(10\), is always less than or equal to the constant congruence speed of their product. Additionally, within the decimal numeral system, we give examples of infinitely many perfect powers whose degree matches their constant congruence speed at every height above \(2\), emphasizing the peculiar recurrence relations of hyper-\(4\). Finally, Appendix B generalizes the decimal framework described above to all squarefree numeral systems, showing that only in such systems the congruence speed stabilizes to a fixed (positive) value for all \(a>1\) not divisible by the radical of the radix. Furthermore, we derive compact formulas for all prime-radix numeral systems and for the composite squarefree senary case.