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Thursday, December 08, 2011

Connecting Diophantine hyperbola to ellipse

The equation x2 - Dy2 = 1 has a direct Diophantine connection to an ellipse for a special case.

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.

So the special case is x = 1 mod D or -1 mod D, which must be true if D is a prime or twice a prime, but the connection will exist whenever x meets the conditions.

Proof:

Using my method to reduce binary quadratic Diophantine equations on x2 - Dy2 = 1, gives:

(x+Dy)2 - D(x+y)2 = -(D-1)

(If you multiply out and simplify you simply return to x2 - Dy2 = 1.)

Shifting things around a bit, I have:

(x+Dy)2 + (D-1) = D(x+y)2

And I can group to divide off D:

[(x+Dy)2 - 1]/D + 1 = (x+y)2

Now we'll shift to the special case of x = 1 mod D or x = - 1 mod D.

Introducing integer j, I have x+Dy = jD +/- 1, and only one of those cases will be true. That usage is to keep from writing x+Dy = jD + 1 or x+Dy = jD - 1, over and over again.

And making that substitution gives:

[(jD +/- 1)2 - 1]/D + 1 = (x+y)2

And squaring and simplifying a bit gives:

Dj2 +/- 2j + 1 = (x+y)2

So now I just subtract and add j2 so that I can complete the square to get:

(D-1)j2 + (j +/- 1)2 = (x+y)2

And you have your ellipse.

Proof complete.

Here is an example of the result with the famous case of D=61:

17663190492 - 61(226153980)2 = 1

x = 1766319049, y = 226153980, 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

If you'd like to continue further, additional research shows you can now generate yet another hyperbola from the ellipse.


James Harris
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