On the Distribution of the First Point of Coalescence for some Collatz Trajectories

TL;DR

This study numerically investigates the coalescence points in Collatz trajectories for specific classes of integers, revealing their distribution's relation to exponential Diophantine equations and estimating their expected values.

Contribution

It introduces a sophisticated algorithm to analyze coalescence points across different modulus classes and links their distribution to solutions of exponential Diophantine equations.

Findings

Distribution of coalescence points relates to exponential Diophantine solutions

Expected coalescence point for n and 3n+2 is approximately 4/5 of n

Deviations observed at peak estimation of the expected value

Abstract

This paper is a numerical evaluation of some trajectories of the Collatz function. Specifically, I assess the coalescence points of each integer $n\equiv 0 (\bmod{2})$ and $n\equiv 2(\bmod{3})$ through a sophisticated algorithm that has been developed to test on any different modulus classes. The data discovered illustrate that the distribution of the first point of coalescence is closely related to the solutions of some exponential diophantine equation. Afterwards, I show that the first point of coalescence of the integers $n$ and $3n+2$ appear to tend to an expected value of $4/5n$. When the algorithm was pushed to its peak estimation it has been discovered that the expected value begins to deviate from the initial estimation of $4/5n$. The first point of coalescence of the integers $n$ and $3n+2$ appear eradicate from a "step by step" point of view but from a topological point of view seem to be localized around the diophantine solution of some particular functions.