math.NT — Number Theory
Nonconstant congruence speeds of integer tetration: period spectra, prime bases, explicit bounds, and densities
Building on earlier human research into the constant congruence speed of integer tetration, including Appendix B of Ripà and Di Pietro (2026), and on a lifting-the-exponent (LTE) criterion publicly presented by Alper Ferudun for OEIS A390535 and A390576, this model-generated note proposes an explicit account of nonconstant speeds in non-squarefree numeral systems. Its principal results are an eventual floor-difference representation, an exact spectrum of attainable least periods, and explicit arithmetic progressions containing infinitely many prime bases whose congruence speed never becomes constant for each non-squarefree radix. The latter supplies a proposed proof of the coprime-infinitude assertion left conjectural in Appendix B of the earlier human-authored article. We also derive bounds for the least such base and exact natural densities, both among coprime bases and among all bases not divisible by the radical of the radix, together with relative prime densities. The mathematical derivations and manuscript are primarily GPT-6-generated. Ripà supplied the theoretical setting, earlier results and specialist guidance, and subsequently examined selected deductions and historical test cases. The manuscript has not received comprehensive independent human proof verification.