A General Class of Collatz Sequence and Ruin Problem

TL;DR

This paper proves the probabilistic convergence of the Collatz sequence to unity, introduces a generalized Collatz sequence with broader convergence properties, and models its behavior as a ruin problem, revealing new statistical insights.

Contribution

It presents a generalized Collatz sequence framework, proves convergence properties, and models the sequence as a ruin problem, extending understanding of its statistical behavior.

Findings

Probabilistic convergence of Collatz to unity.

Generalized Collatz sequences can converge to integers other than one.

Average stopping time grows linearly with number of digits.

Abstract

In this paper we show the probabilistic convergence of the original Collatz (3n + 1) (or Hotpo) sequence to unity. A generalized form of the Collatz sequence (GCS) is proposed subsequently. Unlike Hotpo, an instance of a GCS can converge to integers other than unity. A GCS can be generated using the concept of an abstract machine performing arithmetic operations on different numerical bases. Original Collatz sequence is then proved to be a special case of GCS on base 2. The stopping time of GCS sequences is shown to possess remarkable statistical behavior. We conjecture that the Collatz convergence elicits existence of attractor points in digital chaos generated by arithmetic operations on numbers. We also model Collatz convergence as a classical ruin problem on the digits of a number in a base in which the abstract machine is computing and establish its statistical behavior. Finally an average bound on the stopping time of the sequence is established that grows linearly with the number of digits.