The Search for Non-Integer Worlds
When you chat with mathematicians about the Collatz Conjecture, there is a common little assumption thrown around that I've heard many times. People say that the root cause of the mysterious patterns in the Stopping Times and Max Value Graphs has to do with prime factorization. Take a number n (that you're about to toss into the x3+1 Collatz system). Prime factorize it. You get
n = p1^a1 * p2^a2 * ... * pr ^ar. If n is even, you perform a ÷2 step and the prime factorization does not change very much. However, if n is odd, you perform a x3+1 step and the prime factorization changes entirely! Humanity currently knows very little, ultimately, about the relationship between the prime factorization of p1^a1 * p2^a2 * ... * pr ^ar and the prime factorization of 3 * (p1^a1 * p2^a2 * ... * pr ^ar ) + 1 = q1^b1 * q2^b2 * ... * qs ^bs. People often say that before we can solve the Collatz Conjecture, we will need a much better understanding of this phenomenon. One such example of this hypothesis is found here on math stock exchange.
I don't know why, but somehow, intuitively, I just felt that this was wrong. I felt like the strange patterns of the Collatz System were much more intimately related to this problem and its mechanics and had very little to do with prime numbers or even of properties of the integers.
I began playing around in my journals with the idea of making non-integer Collatz Worlds, something like a x3.5+1 world. If the x3.5+1 world still contained the mysterious patterns, then we would have disproven the above assertion. We would have established that the patterns of the Collatz Conjecture are not the result of some kind of Number Theoretic properties, as is commonly believed. I thought a lot about this and really wanted to know if it was true or not...
In addition to helping answer this question, non-integer Collatz Worlds would serve us another important purpose; something fascinating! If we could give a tangible meaning to the x3.5+1 world, then we could examine worlds along the entire interval between x3+1 and x5+1 and look at exactly when and how divergence first appears! We could get a solid handle on "how close" x3+1 is to having divergence!