To kick things off: what is this mystical Collatz function? Well, let me tell you, it’s a mathematical function that has been puzzling mathematicians since the 1930s. It goes like this take any positive integer and divide it by two if it’s even or multiply it by three and add one if it’s odd. Then repeat this process with the resulting number until you get to 1.

Now, convergence classes. In math-speak, a convergence class is a set of functions that have similar properties when it comes to converging towards some value or limit. And guess what? The Collatz function has its very own convergence classes! Who knew?!

The first convergence class we’ll discuss is the “I’m never gonna give you up” class, also known as the “Collatz conjecture”. This class consists of all positive integers that follow the Collatz function and eventually lead to 1. The catch? Nobody has been able to prove whether or not this convergence actually happens for every number in this class!

Next up is the “I’m still loving you” class, which includes those numbers that converge towards a cycle of values before reaching 1. For example, if we start with the number 3 and follow the Collatz function, we get: 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. This is known as a “cycle” because it repeats itself infinitely (or at least until you run out of patience).

Now the “I swear I’m gonna make it up to you” class, which includes those numbers that converge towards infinity instead of 1. For example, if we start with the number 256 and follow the Collatz function, we get: 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> 2 -> … This is known as a “divergent cycle” because it repeats itself infinitely (or at least until you run out of patience).

Finally, we have the “I’m sorry for everything that I’ve done” class, which includes those numbers that don’t converge towards anything in particular. For example, if we start with the number 1024 and follow the Collatz function, we get: 1024 -> 512 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> … This is known as a “divergent sequence” because it doesn’t have any clear limit or endpoint.

Who knew that such a simple-looking mathematical function could lead to so much excitement and intrigue? But hey, math is like that sometimes…