I hadn't really thought about this before, but if at most it takes O(log(c's length in bits)), that implies that we're doing some kind of logarithmic operation on the bit string of c, and that in some cases we'll need to iterate the logarithm of the whole bit string. The bigger factorization process is just meant to be a calculation, and not involve iterating through possible a and b values, so that refers to all of these pieces of information that we've been figuring out. We're using the tree to figure out information related to c and the parity of its respective variables.We're then jumping to figuring out the triangular numbers that make up the d+n and x+n squares based on their parity. We're using the specific numbers we find from doing those things to calculate a and b. If it's O(log(c's length in bits)), meaning at some point we need to do a logarithmic iteration, but to actually find the factors we're doing a calculation that just involves plugging in some specific numbers, doesn't that imply that rather than being part of finding the factors, the logarithmic iteration is actually done on the bit string version of c? That would mean one of those specific numbers we're using to calculate a and b has something to do with the powers of 2 that make up c, wouldn't it?
>pre-existing virtual "data structure"
That's what a lot of us have assumed this whole time. It's just a matter of even comprehending what that means. I'm currently trying to see if there's any correlation between binary c values and the binary of the triangle numbers used in x+n squares when x+n is odd and n is even.
Not seeing anything so far. I've sort of picked the variables I've looked at arbitrarily, and I don't even know what we're looking for, so I dunno.