Collatz

654 days ago by calkin

Questions concerning the Collatz conjecture, either for $f(n)$ or $g(n)$ (where $g(n) =(3n+1)/2$ if $n$ is odd.

\begin{enumerate}

\item Do we always reach 4,2,1

\item Is there any relationship between the champions?

\item What happens for other functions instead of 3n+1?

\item How fast do the entries in each level increase?  Clearly the maximum is $2^l$?  How fast does the minimum grow?  How does this relate to champions?

\item How fast do the sizes of the levels grow?  

\item Does the symmetry of the tree continue?

\item Can we extend the functions $f,g$ to the complex plane and ask about complex dynamics?

\item If h(n) is time to hit 1, can we identify the patterns we see in the graph?  Can we do this automatically?  Can we express $h(n)$ as a function?

\item Are there patterns regarding final digits 1,3,5,7,9?

\item What happens for odd numbers?  Are there lots of cycles (-5, for example)?  Are there tracks off to $ -\infty$?

\item Are there any interesting physical interpretations?  Probably not.  Is there an interesting mathematical interpretation?  Quite possibly.

\item What are good ways to code the 3n+1 problem?  What data structures work best to avoid over computing?

\item Are there interesting patterns in the statistics for levels?

\item For $g(n)$, we have upsteps when n is odd, downsteps when n is even.  Clearly if $n$ is chosen uniformly in $\{1, 2, \dots, N\}$ when $N$ is big, the probability of $U$ is about equal to probability of $D$. What are the probabilities of $UU$, $UD$, $DU$, $DD$?  Strings of three or more? 

\item What are the up-down patterns of champions like?   How to visualize the UD pattern?  What is the UD pattern for 27?

\item Champions are inputs for which $h(n)$ is bigger than any earlier value: uber-champions are the points on the convex hull of the graph.  Which champions are uber-champions.  How does this change if we change the plot scaling, perhaps plotting $h(n)$ against $\log(n)$?  Or $\log(h(n))$ against $n$?

\end{enumerate}