(I'm a rank hobbyist and I doubt my input is especially inspired, and I really just want to find out how useful the pattern I noticed is. Apologies aside:)

I was looking at the part of the hyperbolic curve near the middle where the curve is very nearly diagonal, and noticed a simple, exploitable pattern:

http://media.tumblr.com/2ecc00039204...acK1r3qp3f.png
(You'll see that I chose to plot y going downwards, like a times-table.)

Near the centre of the curve those squares go diagonally for some long run and then take one step downwards. This is because the curve is slightly greater than 45

^{o}, obviously.

And if (x+1)*(y-1) < target, then x*y must also be less than target. You are only interested in finding and evaluating where it goes down one.

So you can travel those diagonals using a binary search or by division.

Here is my blog post about this:

http://williamedwardscoder.tumblr.co...g-big-integers
I'm curious as to if others have already noticed this, or exploited this?