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Topic: Pell type equations
Replies: 29   Last Post: Apr 29, 2012 5:37 PM

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Helmut Richter

Posts: 164
Registered: 7/4/06
Re: Pell type equations
Posted: Apr 28, 2012 3:33 PM
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On Sat, 28 Apr 2012, Terry M wrote:

> > An equation of the form x^2 - dy^2 = c
> > (where d is not a perfect square)
> > may have no solution;
> > but if you can find one you can find an infinite number
> > by combining this solution with the general solution
> > of Pell's equation x^2 - dy^2 = 1
> > (which always has an infinity of solutions),
> > in the way I suggested.
> >
> > The equation x^2 - dy^2 = c has a solution
> > if it has a solution modulo 8d, I think.
> >

>
> Do you have any idea as to where I can find out more about this?


For the elementary treatment of such equations see also Chrystal's book
http://djm.cc/library/Algebra_Elementary_Text-Book_Part_II_Chrystal_edited02.pdf
on page 478ff where you need some terminology introduced in the preceding
chapters on continued fractions.

It is from another era when the actual handling of equations was the aim
of the algebraists, and not so much the structure of the solution.

Do you know Dario Alpern's calculator which does not only compute the
solutions but also explains the steps taken? See
http://www.alpertron.com.ar/QUAD.HTM

--
Helmut Richter



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