Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices
Abstract
We further develop the theory of congruence speed for integer tetration in non-squarefree numeral systems, building on Appendix B of [M. Ripà and G. Di Pietro (2026), A compact notation for peculiar properties characterizing integer tetration, Notes on Number Theory and Discrete Mathematics, 32(3), 477-496]. For bases coprime to the radix, we establish an eventual floor-difference representation and determine the exact spectrum of attainable least periods. We construct explicit arithmetic progressions containing infinitely many prime bases with nonconstant congruence speed for every non-squarefree radix, thereby proving the coprime-infinitude conjecture posed in Appendix B. We also derive a uniform quadratic bound for the least coprime base with nonconstant congruence speed and establish exact natural densities of nonstabilizing bases, both among integers coprime to the radix and among those not divisible by its radical, together with the corresponding relative density among primes. Finally, we give necessary and sufficient conditions for the perfect-power congruence-speed identity (Equation (2.12) of the cited paper) to hold in arbitrary radices, determine the exact least eventual period when stabilization fails, and establish a separate criterion for the boundary case of the original parameter restriction.
Related
- Distribution of squarefree integers with double congruence conditions
- A compact notation for peculiar properties characterizing integer tetration
- Nonconstant weakly harmonic maps with constant trace
- A Tetration Distribution
- The exact number of nonconstant positive solutions to the planar isotropic Lp dual Minkowski problem
BibTeX
@misc{ripa2026nonconstant,
title = {{Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices}},
author = {Marco Ripà},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10018},
eprint = {2611.10018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10018},
}
RIS
TY - UNPB
TI - Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices
AU - Marco Ripà
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.10018
UR - https://arxiv.org/abs/2611.10018
AB - We further develop the theory of congruence speed for integer tetration in non-squarefree numeral systems, building on Appendix B of [M. Ripà and G. Di Pietro (2026), A compact notation for peculiar properties characterizing integer tetration, Notes on Number Theory and Discrete Mathematics, 32(3), 477-496]. For bases coprime to the radix, we establish an eventual floor-difference representation and determine the exact spectrum of attainable least periods. We construct explicit arithmetic progressions containing infinitely many prime bases with nonconstant congruence speed for every non-squarefree radix, thereby proving the coprime-infinitude conjecture posed in Appendix B. We also derive a uniform quadratic bound for the least coprime base with nonconstant congruence speed and establish exact natural densities of nonstabilizing bases, both among integers coprime to the radix and among those not divisible by its radical, together with the corresponding relative density among primes. Finally, we give necessary and sufficient conditions for the perfect-power congruence-speed identity (Equation (2.12) of the cited paper) to hold in arbitrary radices, determine the exact least eventual period when stabilization fails, and establish a separate criterion for the boundary case of the original parameter restriction.
ER -