p-Regularity of p-Adic Valuations of Number Theoretic Sequences

When studying the convergence properties of a sequence in the p-adic field of numbers one conventionally adopts arguments from standard p-adic algebra and p-adic analysis. This book introduces a novel approach: that of using the notion of p-regularity (for a fixed prime p) to determine the limiting behavior of p-adic sequences. p-regularity derives from the notion of k-regularity (for some positive integer k) the central construct at the intersection of Automata Theory and Number Theory. The chosen sequences of study in this book are linear recurrent sequences which are generated by analytic p-adic functions and the special factorial sequence of S (a union of congruence classes modulo powers of p) a sequence defined by Dr. Manjul Bhargava.