The 3x + 1 Problem, or the Collatz Conjecture, was originally developed in the early 1930's. It has remained unsolved for over eighty years. Throughout its history, traditional methods of mathematical problem solving have only succeeded in proving heuristic properties of the mapping. Because the problem has proven to be so difficult to solve, many think it might be undecidable. In this paper we brie y follow the history of the 3x + 1 problem from its creation in the 1930's to the modern day. Its history is tied into the development of the Cosper Algorithm, which maps binary sequences into integer families. The Then we discuss the pseudo-code which the Cosper Algorithm is based upon. A simple example is provided to demonstrate the Cosper Algorithm. Afterwards, the generalized 3x + k problem is considered yielding two definitions: k-dependent and k-independent cycles. Finally, some images are provided of various k-dependent cycles.

Author Bio

Benjamin Bairrington graduated with honors from the University of California, Davis in the year 2015 with a bachelors of science in Applied Physics and bachelors of science in Mathematics. He has a passion for writing, cartooning, and drawing. He is currently living in Foshan, China teaching English. He plans to return to graduate school within the next two years to study Mathematics at either University of Beijing, Fudan University, or the National University of Singapore.

Aaron Okano is a recent graduate of the University of California, Davis holding a masters in Computer Science.