> For the complete documentation index, see [llms.txt](https://erhany96.gitbook.io/collatz-prefixes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://erhany96.gitbook.io/collatz-prefixes/theory/path-indexed-prefix-tree/parent-and-child.md).

# Parent & Child

A possible proof-by-contradiction opportunity

Recall that there are two cases that disprove the conjecture:

* A number is **looping**: there is a number $$n = F^j(n)$$ such that $$n\ne1$$ and $$j > 0$$.
* A number is **diverging**: there is a number $$n < F^j(n)$$ for $$j > 0$$.

Now, we can argue as follows:

* The root number in a PIPTree is always a power of two, which is trivially known to **converge** (i.e. reach 1).
* If we can prove that **whatever behavior a parent node has, the children will have the same**; then, we come to conclusion that all numbers must have the same behavior with powers of two, which are known to converge.

## Right Child

We know that for a parent with number $$n$$, the right child has $$n/2$$. It is trivial to see that these will have the same behavior, as the right child is just the result of a single iteration of Collatz function $$F(n)$$.

## Left Child

This is a tricky case. For a parent with number $$n$$, the left child has $$n/2+r$$ where $$r$$ is the root number. We know from the previous section the terminating numbers:

* **Good parent**

$$
g\_L' = 3^{m+1} + \frac{3n\_m + 1}{2^{p\_r + 1 - p\_m}}
$$

$$
g\_R' = n\_m
$$

* **Bad parent**

$$
b\_L' = n\_m + 3^m2^{p\_r+1-p\_m}
$$

$$
b\_R' = \frac{3n\_m+1}{2^{p\_r+1-p\_m}}
$$

{% hint style="warning" %}
Connecting the left child to either the good parent or the right child remains **open research**!
{% endhint %}