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Wednesday, February 01, 2017

When connections can fascinate

Interest from others has often intrigued me and on this blog my find of a deep connection between hyperbolas and ellipses is an area that keeps trending.

Like I see a post where I discussed much in 2010, which stays high in list of popular posts. Back then called what I now call the two conics equation "Pell's equation" which is one of those weird to me things described as a historical mistake. Problem was some guy named Pell didn't deserve credit? Then if is known, why keep giving to him? I followed convention for a while then stopped.

For me my own results are too familiar so it helps to see what resonates with others. It's like, on any given day I can, if I wish, consider something I've discovered, but how do I choose? Looking at what interests others gives me a place to consider.

So I discovered the connection and gave some examples.

The connection is from x2 - Dy2 = 1, with D positive so you have hyperbolas to ellipses where it helps to see an example.

Like consider: 172 - 2(12)2 = 1

and I have this key variable j, which is explained in the referenced post:

j = ((17+2(12)-1)/2 = 20 is a solution giving:

202 + 212 = 292

And I notice that if D-1 is a perfect square relationship found connects to circles but otherwise gives non-circular ellipses.

Also proved you can always find integer solutions for af2 + g2 = h2 by using a rational solution to what I NOW call the two conics equation. In that post you see the other usage all over the place. And that is far as I can see one of the most consistently popular subjects for YEARS on this blog.

But would be curious of any other situation where a rational engine drives integer and therefore Diophantine solutions. So yeah lots to ponder there. If you know of any other cases feel free to comment.

Read referenced post for more, and also there are other posts with label tce, and that label can click below.


James Harris
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