More Jabber about the Collatz Conjecture and a Closed Form for Detecting Cycles on Special Subsequences [Assertion: Collatz cycles]

TL;DR

This paper investigates special subsequences in the Collatz conjecture, deriving a closed-form equation to identify seed values in cycles, which are rare and occur among very large numbers, making them hard to solve.

Contribution

It introduces an implicit mixed integer equation in closed form to detect seed values in Collatz cycles, advancing understanding of the conjecture's cycle structure.

Findings

Cycles occur among extremely large numbers.

A closed-form equation for seed values was derived.

Numerical solving of the equation is challenging due to size.

Abstract

Professor Cadogan at the University of the West Indies identified special starting points that yield long subsequences where the normalization constant, k, is always one. I studied these special sequences and found an implicit mixed integer equation in closed form which if solved would produce seed values in cycling subsequences. Such cycles only occur among extremely large numbers, causing the equation to be difficult to solve numerically.