On the Fundamentals of Collatz Conjecture

Abstract

We differentiate even and odd numbers into various groups and subgroups. We provide the properties of the forms of numbers which fall into each groups and subgroups. We expound on the relationship of a special group of even numbers and the collatz conjecture, we also derive an accurate formula to calculate the steps involved when an even number of the group is the initial value of the collatz operation. For each group and subgroup of odd and even numbers, we discuss the observed pattern of their sequences and also derive accurate formulas for each sequence. Throughout, b, d, k, N, n, x, m, and z all denote positive integers, with d, and N denoting odd numbers, x and z denoting even numbers, and b denoting special even-even numbers The order of priority of the properties of each group is key in the differentiation of the numbers into their various groups and subgroups.