Shot in the dark, but when I was seeing if binary search would work, I found that there were relationships between c and several other variables (I'm pretty sure n was one of those but I'll have to check) that showed you could calculate the maximum and minimum possible values. For example, in a grid of c*i, there were a bunch of linear lines, and you could use the gradient of the lowest (visually lowest) line to find the maximum value and the highest to find the minimum. If n0 is between n and (x+n), and we can find the maximum possible n for a given c and the minimum possible (x+n) (and if they happen to magically not overlap), we'll know the range of possible n0 values. Definitely don't hold your breath, because this probably isn't going to work, but I'll post results.
The gradient thing was again a dead end. It seems like it could be immeasurably useful but probably only in one specific context that possibly isn't relevant to anything we're doing so we might not stumble upon it. We can definitely calculate the only possible n values for a given c (e.g. if c=5543, the maximum possible n is 922 based on the lowest possible a being 3), but obviously that isn't useful.
At least in this example, you could rearrange the (x+n)(x+n) square so that n^2 -1 + f = (n+1)^2. Is this always the case? I should probably stop thinking out loud like this since it usually doesn't work out, but if this is true for all odd (x+n) squares, it means n^2 scales upwards with f.
Bump
Firstly, it's ironic that you would post information about clown-tier identification methods in the form of pdfs. Secondly, if we can't share any code at all without possibly being identified, even if it doesn't have someone's name (let alone their IP or whatever in Teach's case) on it, we can't really collaborate. We're all posting combinations of words that could be linked back to us in the form of posts, too.
You might want to fix the second equation.