On Collatz Conjecture

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.