Chapter 3.3:
My Own System for
Non-Integer Worlds
My goal was to give a tangible meaning to the phrase "the x3.5+1 Collatz World". How should we go about this? Note that the method I'm about to describe is radically different from the traditional Complex Collatz function described on the previous page.
Suppose we start, simply enough, by letting Cx3.5+1(n) be equal to 3.5n+1 when n is odd and n/2 when n is even. The problem with this is that this does not return integer outputs. Ok.
Suppose we instead let Cx3.5+1(n) be equal to (3.5n+1 rounded down to the nearest integer) when n is odd and n/2 when n is even. This is kindof interesting. Let's try it out, maybe with n=7.
7 is odd, so we multiply it by 3.5 and add 1 getting 25.5 which we round down to 25.
25 is odd, so we multiply it by 3.5 and add 1 getting 88.5 which we round down to 25.
88 is even, so we divide by 2 getting 44.
The orbit looks something like:
7 >> 25 >> 88 >> 44 >> 22 >> 11 >> 39 >> 137 >> 480 >> 240 >> 120 >> 60 >> 30 >> 15 >>
>> 53 >> 186 >> 93 >> 326 >> 163 >> 571 >> 1999 >> 6997 >> etc...
While this is fun and kind of new and different, there is still a fundamental difference between this idea and traditional Collatz systems... In the classical x3+1 world, every time you take an odd multiply by 3 and add 1 step, you universally end up getting back an even number. This is not true in the new system. One of the core distinctive features of Collatz hailstones is that they only ever fly up by exactly one step at a time... and then plummet down a slightly unpredictable number of times. Okay. Suppose we remedy that.
Suppose we instead let Cx3.5+1(n) be equal to (3.5n+1 rounded down to the nearest even integer) when n is odd and n/2 when n is even. This should solve the problem. Let's try it out with n=7 again.
7 is odd, so we multiply it by 3.5 and add 1 getting 25.5 which we round down to the nearest even number below it, 24.
24 is even, so we divide by 2 getting 12.
12 is even, so we divide by 2 getting 6.
6 is even, so we divide by 2 getting 3.
3 is odd, so we multiply it by 3.5 and add 1 getting 11.5 which we round down to the nearest even number below it, 10.
The orbit looks something like:
7 >> 24 >> 12 >> 6 >> 3 >> 10 >> 5 >> 18 >> 9 >> 32 >> 16 >> 8 >> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
>> 4 >> 2 >> 1 >>
etc...
Nice! This way of defining a non-integer Collatz system seems to have all the properties we desired. Everything seems fairly familiar here, even down to our accustomed 4 >> 2 >> 1 loop. We seem to have finally found what we were looking for. With this new system, we should be able to explore both the how and when of how divergence first appears between x3+1 and x5+1 as well as to test the common assumption that the patterns of the Stopping Times and Max Value Graphs have something to do with properties of prime factorizations.
For the sake of clarity, let me formally define non-integer Collatz Worlds:
We now have everything we need.
