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 such that and .
A number is diverging: there is a number for .
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 , the right child has . 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 .
Left Child
This is a tricky case. For a parent with number , the left child has where is the root number. We know from the previous section the terminating numbers:
Good parent
Bad parent
Connecting the left child to either the good parent or the right child remains open research!
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