Tuesday, January 20, 2009
JSH: Pell's Equation, circle, ellipses, nifty little result
Oh yeah, a while back I found I could get Pythagorean triples from x^2 - Dy^2 = 1, which I thought was neat, but I ended up moving to other things. Recently, however, a poster brought the subject up in a somewhat different context, which got me to thinking about it again, so here's the result:
Given x^2 - Dy^2 = 1
you have solutions for an ellipse or Pythagorean triples with
(D-1)j^2 + (j+/-1)^2 = (x+y)^2
where j = ((x+Dy) -/+1)/D.
Notice that an integer j will always exist as x = +/-1 mod D, from Pell's Equation.
And that's it. I think it's nifty. Tiny. Concise. Does the job.
You can also at times use it to go BACKWARDS and get a solution to Pell's Equation from Pythagorean triples.
And it gives one other result which is that in general, with the Diophantine equation:
ax^2 + y^2 = z^2
where 'a' is a natural number there are always integer solutions and always an infinity of them, driven, I think intriguingly, by solutions to Pell's Equation.
So it's like, discrete ellipses and circles connected to discrete hyperbolas. You could kind of like call it, discrete conic love.
Given x^2 - Dy^2 = 1
you have solutions for an ellipse or Pythagorean triples with
(D-1)j^2 + (j+/-1)^2 = (x+y)^2
where j = ((x+Dy) -/+1)/D.
Notice that an integer j will always exist as x = +/-1 mod D, from Pell's Equation.
And that's it. I think it's nifty. Tiny. Concise. Does the job.
You can also at times use it to go BACKWARDS and get a solution to Pell's Equation from Pythagorean triples.
And it gives one other result which is that in general, with the Diophantine equation:
ax^2 + y^2 = z^2
where 'a' is a natural number there are always integer solutions and always an infinity of them, driven, I think intriguingly, by solutions to Pell's Equation.
So it's like, discrete ellipses and circles connected to discrete hyperbolas. You could kind of like call it, discrete conic love.
# posted by James Harris @ 01:07 
