2019/11/21 by Mario Weitzer, Weitzer, Mario
Computer Science · Mathematics · Social Sciences · #11C08 (Secondary) #11S82 (Primary) 11A63 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Misinformation and Its Impacts #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1911.09624
openalex publication_date 2019/11/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
This article introduces a new kind of number systems on p-adic integers\nwhich is inspired by the well-known 3n+1 conjecture of Lothar Collatz. A\np-adic system is a piecewise function on \ℤp which has branches\nfor all residue classes modulo p and whose dynamics can be used to define\ndigit expansions of p-adic integers which respect congruency modulo powers of\np and admit a distinctive "block structure". p-adic systems generalize\nseveral notions related to p-adic integers such as permutation polynomials\nand put them under a common framework, allowing for results and techniques\nformulated in one setting to be transferred to another. The general framework\nestablished by p-adic systems also provides more natural versions of the\noriginal Collatz conjecture and first results could be achieved in the context.\nA detailed formal introduction to p-adic systems and their different\ninterpretations is given. Several classes of p-adic systems defined by\ndifferent types of functions such as polynomial functions or rational functions\nare characterized and a group structure on the set of all p-adic systems is\nestablished, which altogether provides a variety of concrete examples of\np-adic systems. Furthermore, p-adic systems are used to generalize Hensel's\nLemma on polynomials to general functions on \ℤp, analyze the\noriginal Collatz conjecture in the context of other "linear-polynomial p-adic\nsystems", and to study the relation between "polynomial p-adic systems" and\npermutation polynomials with the aid of "trees of cycles" which encode the\ncycle structure of certain permutations of \ℤp. To outline a\npotential roadmap for future investigations of p-adic systems in many\ndifferent directions, several open questions and problems in relation to\np-adic systems are listed.\n