Title: Aspects of the $3x+1$ Problem
Abstract: The $3x + 1$ problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers $n$ to $3n + 1$ and even integers $n$ to $\frac{n}{2}$. The $3x + 1$ Conjecture asserts that, starting from any positive integer $n$, repeated iteration of this function eventually produces the value 1. The $3x + 1$ Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos commented concerning the intractability of the $3x + 1$ problem: "Mathematics is not yet ready for such problems." Despite this doleful pronouncement, study of the $3x + 1$ problem has not been without reward. It has interesting relations with questions of ergodic theory on the 2-adic integers, and with computability theory. I'll share some of its rich history, some approaches and results of serious researchers, and mention some modest results in studying this delightfully, intricate problem.