This is an unsolved problem in number theory and discrete dynamical systems. There have been hundreds of papers written on this topic, and numerous extensive computations. The field is still fertile, and a good idea could establish something useful. It is a rich ground for computational analysis in order to help gain insight.
Think about the function of a natural number variable defined as follows:
We have and .
The iterates of are defined as follows: , so for all natural numbers .
The –orbit of a natural number is the set .
It is known that for all the T-orbit of eventually contains 1; that is, for some .
The Collatz conjecture is that for all natural numbers there is a natural number (dependent on ) for which .
- Collatz conjecture (and references – particularly those of Lagaris).
Jean Paul Van Bendegem (2005). The Collatz conjecture. A case study in mathematical problem solving. Logic and Logical Philosophy, Vol 14, 7–23 (jpvb_collatz)
The starting point for this investigation is the fact that the Collatz function defined above, can be extended to a a complex-valued function of a complex variable as follows: . We will firstly look at as a function of a real variable .
You can see that this extension of the Collatz function to real number variables is of the form where . We could take the function to be piecewise linear, which would make the analysis of the iterates of the extended form of somewhat easier for a real variable, but would not give us an analytic (= differentiable) function of a complex variable .
With this extended understanding of the function T we see that T does not map the unit interval into itself because . Does T map the interval into itself? No:
The maximum value of occurs at .
The function T maps the interval into itself:
The first few iterates of T look as follows:
Investigate the long term behavior of the iterates of T on .
Complex Collatz function
When we think of as a complex-valued function of a complex variable we can apply it to various subsets of the complex numbers. In particular, when we apply the Collatz function to the unit circle . What we get is the following closed curve:
Applying to this curve gives another closed curve that is significantly more blown up:
Applying T to this blown up curve gives an even more blown up curve. As becomes increasingly unbounded (although always a closed curve).
To the contrary, if we take a circle of radius and apply to this circle we obtain a smaller closed curve:
Applying T to this curve we get an even smaller closed curve:
By the time we have applied to the circe of radius 5 times, we have a much smaller closed curve:
As decreases to 0 (although is always a closed curve).
This investigation is to find a such that the iterates of , applied to the circle of radius gives, as , a limiting closed curve of finite non-zero area. Presumably for the iterates applied to the circle of radius converge to 0, while for the iterates applied to the circle of radius produce closed curves of width that goes to infinity.