The Collatz Theorem

The Collatz conjecture is a famous problem in number theory. Given an integer, if it's odd, multiply it by three and add one, or, if it's even, divide it by two. The Collatz conjecture states that any trajectory of iterates of this Collatz transformation on the positive integers will reach one in a finite number of steps. This problem explores the behavior of a complicated discrete dynamical system that has eluded solution for over seventy years. This paper addresses the Collatz conjecture by altering the Collatz transformation into a friendlier format, which tells us what to do with an odd integer given its congruence modulo eight. We then describe how to find the numbers whose first few iterates follow a given pattern, which leads us to a directed graph that every trajectory must eventually enter. This directed graph then shows us that, in a finite number of steps, every iterate of a trajectory must either converge to one or strictly increase thereafter. Since there is no number whose trajectory strictly increases, the Collatz conjecture holds.