Behavior of numbers is endlessly fascinating to me, and math gives an infinity of tools to study them! Helps to consider the simple and well known not surprisingly, and a post from 2009 covers some basic facts about the equation:
x2 - Dy2 = 1
Where traditionally you find integer solutions for it, which goes back hundreds of years, but it's actually trivial to solve for rational solutions:
x = (D + t2)/(D - t2)
y = -2t/(D - t2)
Here are some easy examples. I like easy:
Let D = -11, and t = 1, then: x = -10/-12 = 5/6, y = -2/-12 = 1/6,
And as required: 25/36 + 11(1/36) = 1
Advance to t = 2, then: x = -7/-15 = 7/15, y = -4/-15 = 4/15,
And as required: 49/225 + 11(16/225) = 1
Expressions give a well known parametric equation for the circle with D=-1.
So yeah, you can actually use those equations to graph hyperbolas or ellipses. And D is related to eccentricity, which is a calculation I've not done, though I've seen someone give the expression showing how they relate. Though I do wonder, what if people had seized upon this way, instead of the way with eccentricity to graph?
Those solutions definitely look simpler. And in our age with computers, not hard to code, though haven't coded. So have no idea if is much worse than traditional ways or not. I DO know I don't typically see those equations and only found out they've been known for centuries when got excited when I re-derived them on my own, and then went looking and found out I didn't have an original discovery. Were actually known to Fermat himself.
I sat down and wrote up a bunch of other things about it back then. Like, for any positive integer D, if D+/-2 is a perfect square, then D+/-1 is the first solution.
82 - 7*32 = 1
Where 7+2 = 9, so 8 is the first solution, and 3, is second. Yeah. So was thinking to myself need to focus on numbers again.
That rule works out to infinity so I might as well make one up instead of just using the example from the page. So, um, say 169, which is 132, so D = 167 and 168 should work.
1682 - 167*132 = 1
And that expression x2 - Dy2 = 1 is SO weird in a way in that there are SO many ways to get trivial solutions for x and y as math people usually describe the easy things, as trivial, so math people would only focus on the harder ones! Years ago I just figured out all the rules for any integer ones, which are known as Diophantine solutions. But now it feels like ancient history to me.
So thinking should talk numbers more as can get bogged down in other things, which I realize interest people less.