Recurrence Relation for a Pell EquationDate: 11/09/1999 at 22:18:14 From: Steve Cash Subject: Pell Equations Can you help me with the following: I need to find a recurrence relation for generating solutions to the Pell equation x^2 - 5y^2 = 1. I can't remember much about Pell equations or recurrence relations from previous math courses I have taken and I'm having trouble finding any information. Thank you for your help. Steve Date: 11/10/1999 at 09:04:53 From: Doctor Floor Subject: Re: Pell Equations Hi, Steve, Thanks for writing. First see my answer to this question in our archives: Triangular Numbers That are Perfect Squares http://mathforum.org/dr.math/problems/manuel.9.07.99.html Now, the minimal solution of x^2 - 5y^2 = 1 is (9,4). So the solutions (u,v) of x^2 - 5y^2 = 1 can be found by: u + v*sqrt(5) = (9 + 4*sqrt(5))^n If you have a solution (u,v), then the next one is found in the following way: (u + v*sqrt(5))(9 + 4*sqrt(5)) = 9u + 9v*sqrt(5) + 4u*sqrt(5) + 20v = 9u + 20v + (4u + 9v)*sqrt(5) So the next solution is (9u+20v, 4u+9v). This can be written in matrix multiplication: ( 9 20 ) ( u ) ( 4 9 ) * ( v ) gives the next solution. I hope this helped you out. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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