 mersenneforum.org Looking for solutions to w^2-n*x^2=z^2
 Register FAQ Search Today's Posts Mark Forums Read 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

Nov 2003

1D2416 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 910 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 1,579 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  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post carpetpool carpetpool 2 2017-02-09 06:41 EdH Linux 16 2016-03-18 17:20 flouran Math 20 2009-09-08 05:48 Vijay Math 6 2005-04-14 05:19 Orgasmic Troll Puzzles 12 2003-07-16 09:36

All times are UTC. The time now is 22:29.

Tue Jan 19 22:29:19 UTC 2021 up 47 days, 18:40, 0 users, load averages: 2.24, 2.00, 1.98