Integer solution to an hyperbola equation

Question

Given the general equation of an hyperbola

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $ where $B^2-4AC>0$

is it possible to find all integers solutions $(x,y)$ as a function of $A, B, C, D$ and $ F $ ? Eventually all the coefficient can be integers as well.

In particular i'm looking for the integer solutions of the equation $2x^2-y^2-y=0$

Thanks

Answer 1

Your final equation turns into $(2y+1)^2 - 8 x^2 = 1.$ In the output below, we show how to get all $U_n^2 - 8 V_n^2 = 1,$ where we get $$ U_{n+2} = 6 U_{n+1} - U_n, $$ $$ V_{n+2} = 6 V_{n+1} - V_n. $$ So, let's see, for you, $x_n = V_n.$ Note that $U_n$ is always odd, and $y_n = \frac{U_n - 1}{2}.$ If you want to include the negative values of $y,$ take $2y+1 = -U$ or $y_n' = \frac{-U_n - 1}{2}.$

jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
  Automorphism matrix:  
    3   8
    1   3
  Automorphism backwards:  
    3   -8
    -1   3

  3^2 - 8 1^2 = 1

 U^2 - 8 V^2 = 1

Wed Jul 13 15:06:18 PDT 2016

U:  3  V:  1 ratio: 3  SEED   BACK ONE STEP  1 ,  0
U:  17  V:  6 ratio: 2.83333
U:  99  V:  35 ratio: 2.82857
U:  577  V:  204 ratio: 2.82843
U:  3363  V:  1189 ratio: 2.82843
U:  19601  V:  6930 ratio: 2.82843
U:  114243  V:  40391 ratio: 2.82843
U:  665857  V:  235416 ratio: 2.82843
U:  3880899  V:  1372105 ratio: 2.82843
U:  22619537  V:  7997214 ratio: 2.82843

Wed Jul 13 15:06:58 PDT 2016

 U^2 - 8 V^2 = 1

jagy@phobeusjunior:~$