Research in Scientific Computing in Undergraduate Education

Collatz conjecture

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 T of a natural number variable n defined as follows:

T(n) = \left\{ \begin{array}{ll} n/2 & \textrm{if }n \textrm{ is even}\\ (3n+1)/2& \textrm{if } n \textrm{ is odd}\end{array} \right.

We have T(6)=6/2=3 and T(3)=(3\times3+1)/2=5.

The iterates of T are defined as follows: T^1=T,T^{k+1}=T \circ T^k, so T^{k+1}(n)=T(T^k(n)) for all natural numbers n.

The Torbit of a natural number n is the set \{T^k(n)\vert k=1,2,3,\ldots\}.

It is known that for all n \leq 17\times2^{58} the T-orbit of n eventually contains 1; that is, T^k(n)=1 for some k.

The Collatz conjecture is that for all natural numbers n there is a natural number k (dependent on n) for which T^k(n)=1.

Reading:

  • 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)

Collatz function

The starting point for this investigation is the fact that the Collatz function T defined above, can be extended to a a complex-valued function of a complex variable as follows: T(z) = \frac{1}{4}(1+4z-(1+2z)cos(\pi z)). We will firstly look at T as a function of a real variable z.

You can see that this extension of the Collatz function to real number variables is of the form \frac{1}{4}(1+4z)-(1+2z)f(z)) where f(z)=1\textrm{ if } z \textrm{ is an even integer, and } -1 \textrm{ if } z \textrm{ is an odd integer}. We could take the function f to be piecewise linear, which would make the analysis of the iterates of the extended form of T somewhat easier for a real variable, but would not give us an analytic (= differentiable) function of a complex variable z.

With this extended understanding of the function T we see that T does not map the unit interval \lbrack 0,1\rbrack into itself because T(1)=2 \textrm{ and } T(2)=1. Does T map the interval \lbrack 0,2\rbrack into itself? No:

The maximum value of T(z) \textrm{ for } 0\leq z \leq 2 \textrm{ of } \approx 2.13924 occurs at z^*\approx 1.18094 \textrm{ - a root of } 4-2cos(\pi z)+\pi(1+2z)sin(\pi z)=0.

The function T maps the interval \lbrack 0,z^*\rbrack \approx \lbrack 0,2.13924\rbrack into itself:

The first few iterates of T look as follows:

Collatz function - second iterate on [0,z*]

Collatz function - third iterate on [0,z*]

Collatz function - fourth iterate on [0,z*]

Investigate the long term behavior of the iterates of T on \lbrack 0,z^*\rbrack.

Complex Collatz function

When we think of T 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 T to the unit circle S^1:=\{z \in \mathbb{C}: \vert z\vert=1\}. What we get is the following closed curve:

Applying T 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 k \textrm{ increases } T^k(S^1) becomes increasingly unbounded (although always a closed curve).

To the contrary, if we take a circle S^1(\frac{1}{5}):=\{z\in \mathbb{C}:\vert z\vert=\frac{1}{5}\} of radius \frac{1}{5} and apply T 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 T to the circe of radius \frac{1}{5} 5 times, we have a much smaller closed curve:

As k \textrm{ increases } T^k(S^1(\frac{1}{5})) decreases to 0 (although is always a closed curve).

This investigation is to find a \lambda_0, 0<\lambda_0<1 such that the iterates T^k of T, applied to the circle of radius \lambda_0 gives, as k \to \infty, a limiting closed curve of finite non-zero area. Presumably for \lambda < \lambda_0 the iterates T^k \textrm{ of } T applied to the circle of radius \lambda converge to 0, while for \lambda > \lambda_0 the iterates T^k \textrm{ of } T applied to the circle of radius \lambda produce closed curves of width that goes to infinity.

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