Solving this three variable diophantine equation.

Question

What technique(s) can be employed to solve this equation for integers? My target is non-zero integer solutions. I don't have a clue on how to solve this purely by hand. :

$z^2 = (y^3 - 51)^2 - 12k \cdot (6y^2 - \sqrt{36y^4 - 48y(y^3 - 51) + 576k} )$

This equation is a merger of two equations:

$12x^2 - 6y^2 \cdot x + y(y^3 - 51) - 12k = 0$

and

$z^2 = (y^3 - 51)^2 - 288k \cdot x$

For the first quadratic equation, it's easy to make $k$ the subject of the equation and substitute values into $y$ and work out for $x$ but the results therefrom doesn't fit into the second equation. So I expressed $x$ in terms of the quadratic formula and put into the second equation that's why the initial equation appears monstrous. I'd appreciate it if anyone can find solutions that satisfy both equations.

Thanks!

Answer 1

Resultant $Res_x\bigg(12 x^2 - 6 y^2 x + y (y^3 - 51) - 12 k, -z^2 + (y^3 - 51)^2 - 288 k x\bigg)\\=(z^2 + 72 k y^2 - (-51 + y^3)^2)^2 - 1728 k^2 (48 k + 204 y - y^4)\\ \implies 3(48 k + 204 y - y^4)=\square$

Let $3(48 k + 204 y - y^4)=w^2$.

$Res_k\bigg(\bigg\{12 x^2 - 6 y^2 x + y (y^3 - 51) - 12 k, -z^2 + (y^3 - 51)^2 - 288 k x\bigg\}, 3 (48 k + 204 y - y^4) - w^2\bigg)\\=\bigg\{(w + 12 x - 3 y^2) (w - 12 x + 3 y^2), (-2601 + 2 w^2 x - 1224 x y + 102 y^3 + 6 x y^4 - y^6 + z^2)\bigg\}$

$Res_x\bigg((-2601 + 2 w^2 x - 1224 x y + 102 y^3 + 6 x y^4 - y^6 + z^2), (w + 12 x - 3 y^2)\bigg)\\=(15606 + w^3 - 612 w y - 3 w^2 y^2 + 1224 y^3 + 3 w y^4 - 3 y^6 - 6 z^2)$

Let $y$ parameter, then get elliptic equation over $w,z\in\mathbb{Z}$:

$w^3 - w^2 3 y^2 + w 3 y (-204 + y^3) - 3 y^3 (-408 + y^3) + 15606 = 6 z^2$

Solving in pari/gp:

{
 for(y=-1000, 1000,
  H= hyperellratpoints((15606+'w^3-612*'w*y-3*'w^2*y^2+1224*y^3+3*'w*y^4-3*y^6)/6, 10^7);
  for(i=1, #H,
   z= H[i][2];
   if(z>0, if(z==floor(z),
    w= H[i][1];
    k= (w^2-612*y+3*y^4)/144;
    if(k==floor(k),
     x= (3*y^2-w)/12;
     if(x==floor(x),
      if(x!=0, if(12*x^2-6*y^2*x+y*(y^3-51)-12*k==0 && -z^2+(y^3-51)^2-288*k*x==0,
       print("("x", "y", "z", "k")")
      ))
     )
    )
   ))
  )
 )
}

Output (x,y,z,k) for $|y|<1000$:

(-488917, -984, 8882365635, 553864886575)
(-86359, -939, 1851808374, 110316146278)
(-358020, -489, 4258525020, 175748277075)
(4380, -309, 12316560, 569800770)
(-10104, -309, 69151776, 1344180588)
(4344, -285, 6720168, 392242566)
(-127896, -285, 902562072, 22101356046)
(-12015, -99, 27053970, 211245120)
(-28, -21, 16560, 23254)
(-2716, -21, 2500272, 7991830)
(-1071, 0, 594813, 1147041)
(-1560, 51, 1508520, 5025930)
(-247, 69, 537030, 2537626)
(-6783, 264, 40999395, 687176319)
(11424, 459, 26676672, 2625969600)
(13289, 465, 4544838, 2635992751)
(-122317, 465, 1067827410, 32081547394)