I have a question about the Collatz conjecture and how some numbers merge trajectories.
Take the standard map:
$$C(n) = \begin{cases} n/2, & \text{if $n \equiv 0$ mod $2$} \\ 3n+1, & \text{if $n \equiv 1$ mod $2$} \end{cases}$$
Suppose we have a minimal counterexample to the Collatz conjecture, $k \equiv 1 \text{ mod } 3 \in \Bbb{Z^+}$ (ignoring multiples of $3$ for now). Can it be proven that $k$ must join trajectories with any number of the form $k+3m+1$, such that $6m+1<k$? This last condition may seem a bit strange but it is needed so that smaller values of $k$ will suit this.
For example, if $m = 1$, can it be proven that $k+4$ will merge with $k$, for all, sufficiently large $k \equiv 1 \text{ mod } 3$? If that doesn't work, what about $m=2$?
To sum up, will the trajectory of $k+4$ or $k+7$ etc. ever meet the trajectory of $k$? As always, help will be much appreciated, though it will be unlikely that anything will come from this.