On Collatz Conjecture

TL;DR

This paper introduces new fractional sum notations and formulas to analyze Collatz sequences, providing insights into sequence lengths, loops, and graph structures, advancing understanding of the conjecture's properties.

Contribution

It presents novel fractional sum notations, formulas for sequence lengths, and a Collatz graph generation method, offering new tools for studying the conjecture's dynamics.

Findings

Formula for numbers with sequence length 2

Proof that only trivial 2-cycle exists

Procedure to generate Collatz graph

Abstract

The Collatz Conjecture can be stated as: using the reduced Collatz function $C(n) = (3n+1)/2^x$ where $2^x$ is the largest power of 2 that divides $3n+1$, any odd integer $n$ will eventually reach 1 in $j$ iterations such that $C^j(n) = 1$. In this paper we use reduced Collatz function and reverse reduced Collatz function. We present odd numbers as sum of fractions, which we call `fractional sum notation' and its generalized form `intermediate fractional sum notation', which we use to present a formula to obtain numbers with greater Collatz sequence lengths. We give a formula to obtain numbers with sequence length 2. We show that if trajectory of $n$ is looping and there is an odd number $m$ such that $C^j(m) = 1$, $n$ must be in form $3^j\times2k + 1, k \in \mathbb{N}_0$ where $C^j(n) = n$. We use Intermediate fractional sum notation to show a simpler proof that there are no loops with length 2 other than trivial cycle looping twice. We then work with reverse reduced Collatz function, and present a modified version of it which enables us to determine the result in modulo 6. We present a procedure to generate a Collatz graph using reverse reduced Collatz functions.