Integer patterns in Collatz sequences

TL;DR

This paper constructs a graph based on inverse Collatz functions to explore integer patterns, offering new insights that could contribute to proving the Collatz conjecture.

Contribution

It introduces an arborescence graph derived from inverse Collatz iterations, revealing novel integer patterns relevant to the conjecture's proof.

Findings

New arborescence graph structure from inverse Collatz iterations

Identification of integer patterns in the graph

Potential implications for proving the Collatz conjecture

Abstract

The Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an arborescence graph constructed from iterations of $g(x) = (2^{e(x)}x - 1)/3$, which is the inverse of $f(x)$ and where $x \not \equiv [0]_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x - 1 \equiv [0]_3$, with $[0]_3$ denoting $0\pmod{3}$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity of the conjecture.