Chapter 3.5: Let's Find Out
I revved my Python engines, put the petal to the metal, and pressed go! Here are some of the fascinating results I got back:
The x3.5+1 World
The x3.141592+3.141592 World
The x2.9+11 World
The x1.7+2.5 World
The x3.25+7.5 World
The x3.564+1 World
Well, the data was in. Clearly the mysterious patterns in the x3+1 Collatz World's Stopping Times and Max Value Graphs have nothing at all particularly to do with prime factorization. That common assumption honestly looks to be wrong.
Perhaps more relevantly... Jeeez some of these graphs are gorgeous! I was taken aback and honestly kind of jaw-dropped when I first went exploring in these never-before-seen parts of the Collatz Wilderness. I've posted just a handful of these images above, but in those first days I explored thousands and thousands of Collatz Worlds, every one of them more beautiful and mysterious than the last. If you would like your own shot at exploring out there, skip over to the Python Programs section of this website and download program #2 - Input xa+b, Output the Two Graphs.
In the next chapter we will take a much closer look at these xa+b Collatz Worlds. What are the patterns found across all of these worlds? What knowledge can we learn from studying them? And how does the x3+1 world (from the original Collatz Conjecture) fit into all this?
The next chapter is really cool!