The Curling Number Conjecture

TL;DR

This paper explores the Curling Number Conjecture, which suggests that repeatedly extending a sequence by its curling number will eventually reach 1, with focus on sequences of 2's and 3's.

Contribution

It provides numerical results and conjectures related to the conjecture, especially for sequences of only 2's and 3's.

Findings

Numerical evidence supporting the conjecture for specific sequences.

Formulation of new conjectures based on observed patterns.

Analysis of sequences composed of 2's and 3's.

Abstract

Given a finite nonempty sequence of integers S, by grouping adjacent terms it is always possible to write it, possibly in many ways, as S = X Y^k, where X and Y are sequences and Y is nonempty. Choose the version which maximizes the value of k: this k is the curling number of S. The Curling Number Conjecture is that if one starts with any initial sequence S, and extends it by repeatedly appending the curling number of the current sequence, the sequence will eventually reach 1. The conjecture remains open, but we will report on some numerical results and conjectures in the case when S consists of only 2's and 3's.