What is the benefit of looking at the binary representation of the collatz conjecture. I know that it makes the computation easier because there is really one operation involved which is multiplying by 3. And then we simply shift the resulting binary number to the right if it hast Least significant bit (lsm) = 0
Does this offer a faster algorithm (complexity wise) ?
I know that the normal algorithm for collatz doesn't have an upper bound using Big Oh notation. (It's not analysable).
It is natural to consider (and analyze) the Collatz map not as an operation on numbers but on strings. Most obvious candidates are the strings representing the numbers in bases $2$, $3$, and $6$. In base $6$ the Collatz map works like a cellular automaton, i.e., can be computed for the different digits "in parallel". The reason for this is the following: $\>:\!2$ and $\>\times3$ are the same in base $6$; furthermore the resulting carries have no long-range effects in this base. (Note that for multiplications or divisions in some base you usually have to work from the left or from the right and cannot decide on a particular digit until all digits to the left or to the right of it have been worked out and the remaining carry is determined.)