From my method to generally reduce binary quadratic Diophantine equations, I found the following when using on an already reduced form.
It must be that if you have:
u2 + Dv2 = F
then it must also be true that
(u-Dv)2 + D(u+v)2 = F(D+1)
Which I eventually decided to call a Binary Quadratic Diophantine iterator, and you can do SO much with it. Thought I'd highlight a few things here.
You can find sums of square for the same value with it.
52 + 202 = 17*52
82 + 192 = 17*52
132 + 162 = 17*52
Using it you can get an infinity of expressions to approximate the square root of 3, where posted about one which will show here:
sqrt(3) approximately equals xn+1/yn+1, where:
xn+1 = 362xn + 627yn
yn+1 = 209xn + 362yn
and x0 = 1, and y0 = 0;
Next, x1 = 362 and y1 = 209, and (362/209)2 approximately equals: 3.0000228
Iterate, and you get, x2 = 262087, y2 = 151316
And: (262087/151316)2 approximately equals 3.00000000004367
Iterate again: x3 = 189750626 and y3 = 109552575
(189750626/109552575)2 approximately equals 3.0000000000000000833
And 189750626/109552575 approximately equals 1.732050807568877 showing only digits that match with sqrt(3).
You can find a sum of squares to equal a square with as many as you like.
To demonstrate I posted an example doing 5 sums of squares to get a square.
Where I found, two sums of fives squares that give a square:
42 + 62 + 102 + 142 + 862 = 882
862 + 1292 + 2152 + 3012 + 8812 = 9682
You can do lots of things of that type and I guess I played around quite a bit. Also though found could explain something noticed by Euler and Ramanujan which was cool.
It is interesting that I found a burst of things after coming up with name Binary Quadratic Diophantine iterator or BQD Iterator for short. Why would naming it make such a difference for me?
Am sure I did things with it before naming it as have had it for years now. Like I found could connect ellipses and hyperbolas back before I had the full current form. There's just so MUCH so will just go ahead and stop there. Feel like covered enough things for this post.