An introduction to $p$-adic systems: A new kind of number systems inspired by the Collatz $3n+1$ conjecture

Abstract

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all residue classes modulo $p$ and whose dynamics can be used to define digit expansions of $p$-adic integers which respect congruency modulo powers of $p$ and admit a distinctive "block structure". $p$-adic systems generalize several notions related to $p$-adic integers such as permutation polynomials and put them under a common framework, allowing for results and techniques formulated in one setting to be transferred to another. The general framework established by $p$-adic systems also provides more natural versions of the original Collatz conjecture and first results could be achieved in the context. A detailed formal introduction to $p$-adic systems and their different interpretations is given. Several classes of $p$-adic systems defined by different types of functions such as polynomial functions or rational functions are characterized and a group structure on the set of all $p$-adic systems is established, which altogether provides a variety of concrete examples of $p$-adic systems. Furthermore, $p$-adic systems are used to generalize Hensel's Lemma on polynomials to general functions on $\mathbb{Z}_p$, analyze the original Collatz conjecture in the context of other "linear-polynomial $p$-adic systems", and to study the relation between "polynomial $p$-adic systems" and permutation polynomials with the aid of "trees of cycles" which encode the cycle structure of certain permutations of $\mathbb{Z}_p$. To outline a potential roadmap for future investigations of $p$-adic systems in many different directions, several open questions and problems in relation to $p$-adic systems are listed.