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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

Alessandro Piazza
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  • If any coefficient is irrational, then, except special cases, no solution is possible. If they are rational, is always possible to reduce them to a common denominator, hence to integers. – G Cab Jul 13 '16 at 22:10
  • Thank you for the correction, i wasn't sure about that and you clarified that point. – Alessandro Piazza Jul 14 '16 at 10:05
  • You are wellcome, for the rest Will's answer indicates how to relate your question to the famous "Pell's equation", so not much to add about more than the vast literature on this subject – G Cab Jul 14 '16 at 15:42

1 Answers1

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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:~$
Will Jagy
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  • Many thanks for your answer. By the way, do you know if it is possible to finde a general formula for the equation above in terms of $A, B, C, D$ and $F$? – Alessandro Piazza Jul 14 '16 at 10:08
  • @AlessandroPiazza, the answer (solution) is already quite involved for the hyperbola in its canonical Pell's equation. I do not think you can find compact formulas for general coefficients, unless they combine into integer translations, scaling, and rotation by $90^\circ$ (or even by $45^\circ$ + appropriate scaling). – G Cab Jul 14 '16 at 15:55