mersenneforum.org Looking for solutions to w^2-n*x^2=z^2
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 2005-04-11, 10:31 #1 jtavares   Nov 2004 32 Posts Looking for solutions to w^2-n*x^2=z^2 How can i find solutions (w, x, z) to the following equation: w^2-n*x^2=z^2 Can i use the continued fraction expansion? and how? Is the solution related to the factorization of n?
2005-04-11, 11:37   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

53×61 Posts

Quote:
 Originally Posted by jtavares How can i find solutions (w, x, z) to the following equation: w^2-n*x^2=z^2 Can i use the continued fraction expansion? and how? Is the solution related to the factorization of n?
We will assume n is squarefree (standard assumption; otherwise just do
a change of variables)

For z = 1, solution methods are well known. This is Pell's equation. For z > 1,
there are extensions of the cfrac method. See Henri Cohen's book. I don't
have it handy, so can't look up the exact reference.

 2005-04-11, 18:46 #3 jtavares   Nov 2004 118 Posts No, Henri Cohen's books do not deal with it - "A course in computacional algebraic number theory" and "Advanced topics in computacional number theory". I could only find the solution to x^2+dy^2=p with d>0 and p prime (Cornachia). I am not sure if this could be used to solve w^2-n*x^2=z^2 by transforming it to z^2+n*x^2=w^2. Anyway there sould be a suitable continued fraction aproximation too but i can not find it.
 2005-04-11, 19:25 #4 Citrix     Jun 2003 31378 Posts an easier method w^2-n*x^2=z^2 then w^2-z^2=n*x^2 or (w-z)* (w+z) =n*x^2 factorize n*x^2 to get solutions of w and z. Citrix

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