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Sunday, July 10, 2016

So much from one thing

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 + 6+ 10+ 14862 = 882

862 + 129+ 215+ 301+ 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.


James Harris
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