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 formintermediate 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.