It must be that if you have:

u

^{2}+ Dv

^{2}= 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.

**5**

^{2}+ 20^{2}= 17*5^{2}**8**

^{2}+ 19^{2}= 17*5^{2}

^{}**13**

^{2}+ 16^{2}= 17*5^{2}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

**x**, where:

_{n+1}/y_{n+1}**x**

_{n+1}= 362x_{n}+ 627y_{n}**y**

_{n+1}= 209x_{n}+ 362y_{n}and x

_{0}= 1, and y

_{0}= 0;

Next, x

_{1}= 362 and y

_{1}= 209, and (362/209)

^{2}approximately equals: 3.0000228

Iterate, and you get, x

_{2}= 262087, y

_{2}= 151316

And: (262087/151316)

^{2}approximately equals 3.00000000004367

Iterate again: x

_{3}= 189750626 and y

_{3}= 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:

**4**

^{2}+ 6^{2 }+ 10^{2 }+ 14^{2 }+**86**

^{2}**= 88**

^{2}**86**

^{2}+ 129^{2 }+ 215^{2 }+ 301^{2 }+ 881

^{2}**= 968**

^{2}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.

James Harris

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