Putting n = 0, 1, .... will result in solutions in the four classes. Putting n = 0 in the second one will give (40, 11), which corresponds to a fundamental solution for on of the classes. Putting n = 0 in the fourth one will give (12, 3) which also corresponds to one of the fundamental solution we have found.
Putting n= 0 in the first one and third one does not yield the same fundamental solutions found by diop_pell() (they simplifies to (-40, -11) and (-12, -3) respectively). I used the LMM algorithm discussed in the paper, so sometimes fundamental solutions found by different algorithms for each class may differ (LMM returns minimal positive solutions so this makes sense). Here is the paper, http://www.jpr2718.org/pell.pdf x**2 - 13*y**2 = 27. diop_pell(13, 27) is discussed at the last paragraph of page 14. Regards, Thilina. On Wed, Jul 3, 2013 at 8:50 PM, Ondřej Čertík <[email protected]>wrote: > Thilina, > > On Wed, Jul 3, 2013 at 8:00 AM, Thilina Rathnayake > <[email protected]> wrote: > > > > > > > > Hi Ondrej, > > > > I don't know much about solving recurrence relations on two variables, > but > > the sources > > I am referencing provides the solutions for the recurrences involved with > > Pell equation. > > So, I think that would be enough. > > Absolutely. > > > > > Concerning the solutions of x**2 - 13*y**2 = 27, solutions returned by > > diop_pell > > are: > >>>> diop_pell(13, 27) > > [(220, 61), (40, 11), (768, 213), (12, 3)] > > > > Here is how wolfram alpha represents it: > > http://www.wolframalpha.com/input/?i=x**2+-+13*y^2+%3D+27 > > > > They provide general solutions for each class. > > (Look in the box named "Integer solutions") > > It has 4 solutions, the first one is: > > x = ±1/2 (-40 (649-180 sqrt(13))^n+11 sqrt(13) (649-180 sqrt(13))^n-40 > (649+180 sqrt(13))^n-11 sqrt(13) (649+180 sqrt(13))^n), y = ±1/26 > (-143 (649-180 sqrt(13))^n+40 sqrt(13) (649-180 sqrt(13))^n-143 > (649+180 sqrt(13))^n-40 sqrt(13) (649+180 sqrt(13))^n), n element Z, > n>=0 > > How is this solution mapped to one of the numbers in [(220, 61), (40, > 11), (768, 213), (12, 3)] ? > > Ondrej > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > For more options, visit https://groups.google.com/groups/opt_out. > > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. For more options, visit https://groups.google.com/groups/opt_out.
