The Collatz Conjecture and Integers of the Form

One of the more well-known unsolved problems in number theory is the Collatz (3n + 1) Conjecture. The conjecture states that iterating the map that takes even n ∈ N to n 2 and odd n to 3n+1 2 will eventually yield 1. This paper is an exploration of this conjecture on positive integers of the form 2kb − m and 3kb − 1, and stems from the work of the first author’s Senior Seminar research. We take an elementary approach to prove interesting relationships and patterns in the number of iterations, called the total stopping time, required for integers of the aforementioned forms to reach 1, so that our results and proofs would be accessible to an undergraduate. Our results, then, provide a degree of insight into the Collatz Conjecture.