Extending 1089 attractor to any number of digits and any number of steps

TL;DR

This paper generalizes the 1089 number trick to any number of digits and steps, revealing new dynamical patterns, cycles, and properties of integers through iterative digital reversal processes.

Contribution

It introduces a generalized iterative digital reversal system and characterizes new cycles, diverging behaviors, and the structure of resulting integers beyond the classical 1089 trick.

Findings

Identification of 2-cycles and higher p-cycles in the system.

Discovery of diverging trajectories with 8-cycle rhythm signatures.

Characterization of Papadakis-Webster integers and binary strings.

Abstract

The well-known 1089 trick reflects an amazing trait of digital reversal process and reminisces of a limiting attractor in dynamical systems even though it takes only two steps. It is natural to consider the situations when the number of digits is beyond three as in the original 1089 trick, as well as situations when the number of steps is beyond two. The first part has been mostly done by Webster which we will reproduce. After two steps, the resulting integers are called Papadakis-Webster integers (PWI), which is always divisible by 99, and the resulting quotients consist of only 0's and 1's, which we name Papadakis-Webster binary strings (PWBS). Not all binary strings could be PWBS, and we define the hairpin pairing rule to determine if a binary string is a PWBS. For the second part, we propose a two-option iteration system named iterative digital reversal (IDR) suitably interweaving additions and subtractions. The simplest limiting behavior of IDR is 2-cycles. The elements in an IDR 2-cycle are all composed of repetitions of the 10(9)$_L$89 (L>=0) motif, and are all PWIs. The lower 2-cycle elements after division of 99 belong to the subset of PWBS that are palindromic and consist of 0- and 1-blocks with a minimal length of two. IDR also has higher p-cycles (p=10,12,71) whose elements seem to contain at least one PWI. Another interesting finding about IDR is that it contains non-periodic and diverging trajectories, as the integer values grow to infinity. In these diverging trajectories, while the number of flanking digits around the middle point increases by the iteration, the middle part has an 8-cycle rhythm or signature which has been found in all diverging trajectories. Overall, the generalization of the original 1089 trick in both space and time leads to new patterns in integers and new phenomenology in dynamics.