One of my more popular research areas explores a connection between ellipses and hyperbolas--two of the three conic sections.
The connection starts with the equation x2 - Dy2 = 1.
It is a hyperbola if D is a positive integer, which is the assumption throughout this post.
Rare situation where I did things in rationals, as I'm going to copy a LOT from a prior post and expanding on it a bit:
(D-1)j2 + (j+1)2 = (x+y)2 where j = ((x+Dy) -1)/D
(D-1)j2 + (j-1)2 = (x+y)2 where j = ((x+Dy) +1)/D
And those are ellipses.
And going to give an example where you get all integers. Why that happens is explained further down, when I include more research from a second post on the subject.
With D=2, and x=17, y=12, 172 - 2(12)2 = 1, and using the first one:
j = ((17+2(12)-1)/2 = 20 is a solution giving:
202 + 212 = 292
And it's like, who knew? But now you know that:
172 - 2(12)2 = 1 is connected to: 202 + 212 = 292
Oh yeah, despite being in rationals, noticed this weird thing that the result means that you can always find integer solutions for an equation of the form:
af2 + g2 = h2
using a rational solution for x2 - Dy2 = 1 when a = D - 1. That's just kind of freaky cool.
For more details including links to derivations check out the full post:
Later I focused exclusively on integers and found more rules.
The ellipse is: (D-1)j2 + (j +1)2 = (x+y)2 where j = ((x+Dy)-1)/D, when x = 1 mod D.
Or: (D-1)j2 + (j -1)2 = (x+y)2 where j = ((x+Dy)+1)/D, when x = -1 mod D.
And I used a famous large solution to show an example:
Here is an example of the result with the famous case of D=61:
17663190492 - 61(226153980)2 = 1
and x = 1766319049 = -1 mod 61, so the second equations apply.
(D-1)j2 + (j - 1)2 = (x+y)2, j = (x+Dy+1)/D
Which gives j = 255110030.
60*2551100302 + 2551100292 = 19924730292
That post, which also includes a proof, is:
So I've copied key points from two related posts.
The connection between ellipses and hyperbolas in rationals is a fascinating one to me. It means that one can be used to generate the other and these are infinity connections.
Despite it being a rational connection, or because of it, there is a Diophantine variant, where the residue of x modulo D is important. The importance of that residue modulo D emerges elsewhere as key to the size of the fundamental solution x2 - Dy2 = 1.
So things may be kind of convoluted for us (or is it just me?), but perfectly logical mathematically where these things are just profoundly connected.
It's kind of interesting then that this connection may be HUGE for the conics involved. It links their solutions, so I guess that makes sense.