TITLE: Solve $32x^2 -y^2 = 448$
QUESTION [5 upvotes]: I am trying to find all integer solutions to the following equation:
$$32x^2 - y^2 = 448$$
This is what I have tried so far:
The equation describes a hyperbola, and so I try the usual trick of intersecting the curve with a line of rational slope to find rational solutions first.
Knowing the point (4,8) satisfies the equation, I solve the following system:
$$\left\{ 
\begin{array}{c}
32x^2 - y^2 = 448 \\ 
y = m(x - 4) + 8 \\
\end{array}
\right.$$
After a bunch of algebra, I get:
$$x = \frac{-4m^2+16m-128}{32-m^2}$$
$$y = \frac{8m^2-256m+256}{32-m^2}$$
Finally, substituting $m = \frac{u}{v}$, I get:
$$x = \frac{-4u^2+16uv-128v^2}{32v^2-u^2}$$
$$y = \frac{8u^2-256uv+256v^2}{32v^2-u^2}$$
Cool, with any choice of $u$ and $v$, I get a rational solution.
But since cancelling the denominators does not work, I do not know how to continue to get integer solutions only.
Is this perhaps not the right way to go?
Any help would be much appreciated.

REPLY [2 votes]: Four orbits under
$$  x_{n+2} = 34 x_{n+1} - x_n,  $$
$$  y_{n+2} = 34 y_{n+1} - y_n.  $$
$$   (4,8); \; \; (92,520);  \; \; (3124,17672); \; \; (106124,600328); $$
$$ (8,40); \; \; (256,1448);  \; \; (8696,49192); \; \; (295408,1671080); $$
$$   (16,88); \; \; (536,3032);  \; \; (18208,103000); \; \; (618536,3498968); $$
$$   (44,248); \; \; (1492,8440);  \; \; (50684,286712); \; \; (1721764,9739768); $$
As sometimes happens, these can be combined into two orbits under
$$  x_{n+2} = 6 x_{n+1} - x_n,  $$
$$  y_{n+2} = 6 y_{n+1} - y_n.  $$
$$   (4,8); \; \; (16,88); \; \; (92,520);  \; \; (536,3032);  \; \;  (3124,17672); \; \;(18208,103000); \; \; (106124,600328); \; \;  (618536,3498968);$$
$$ (8,40); \; \;  (44,248); \; \;(256,1448);  \; \;(1492,8440);  \; \;  (8696,49192); \; \; (50684,286712); \; \; (295408,1671080);  \; \;(1721764,9739768); $$
My program calls them $w,v.$
==================================================
jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
  Automorphism matrix:  
    17   96
    3   17
  Automorphism backwards:  
    17   -96
    -3   17

  17^2 - 32 3^2 = 1

 w^2 - 32 v^2 = -448

Wed Oct  4 07:13:21 PDT 2017

w:  8  v:  4 ratio: 2  SEED   KEEP +- 
w:  40  v:  8 ratio: 5  SEED   KEEP +- 
w:  88  v:  16 ratio: 5.5  SEED   BACK ONE STEP  -40 ,  8
w:  248  v:  44 ratio: 5.63636  SEED   BACK ONE STEP  -8 ,  4
w:  520  v:  92 ratio: 5.65217
w:  1448  v:  256 ratio: 5.65625
w:  3032  v:  536 ratio: 5.65672
w:  8440  v:  1492 ratio: 5.65684
w:  17672  v:  3124 ratio: 5.65685
w:  49192  v:  8696 ratio: 5.65685
w:  103000  v:  18208 ratio: 5.65685
w:  286712  v:  50684 ratio: 5.65685
w:  600328  v:  106124 ratio: 5.65685
w:  1671080  v:  295408 ratio: 5.65685
w:  3498968  v:  618536 ratio: 5.65685
w:  9739768  v:  1721764 ratio: 5.65685
w:  20393480  v:  3605092 ratio: 5.65685
w:  56767528  v:  10035176 ratio: 5.65685
w:  118861912  v:  21012016 ratio: 5.65685
w:  330865400  v:  58489292 ratio: 5.65685

Wed Oct  4 07:15:22 PDT 2017

 w^2 - 32 v^2 = -448

jagy@phobeusjunior:~$ 

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