On the integer solutions of $aX^2+bY^2=cZ^2+q$

Question

In this article in AoPS the author gives a parametric solution to

$$ \begin{equation} aX^2+bY^2=cZ^2+q \tag 1 \end{equation} $$

in terms of the solutions to the equation

$$ax^2+by^2=cz^2 \tag 2$$ as

$$ \begin{cases} \begin{align} X=\frac{x}{2}(ck^2-as^2-bp^2+q)+s \\ Y=\frac{y}{2}(ck^2-as^2-bp^2+q)+p \\ Z=\frac{z}{2}(ck^2-as^2-bp^2+q)+k \end{align} \end{cases} \tag 3 $$

where $s,p,k$ are solutions of the linear diophantine equation

$$axs+byp-czk=1. \tag 4$$

Question: Can we generate all solutions of $(1)$ from all solutions of Eqn. $(2)$ or is just a single particular solution of $(2)$ sufficient?

As an example, I was trying to solve $X^2 + 3Y^2 = Z^2 + 144832$. The equation $x^2 + 3y^2 = z^2$ has particular solutions $(1,0,1), (1,1,2), \cdots$.

Are the solutions $(X,Y,Z)$ generated from $(1,0,1)$ equivalent to the solutions generated from $(1,1,2)$?

Answer 1

Here is output as I indicAted, using $u,v$ positive only. One may then negate $x$ or $z$ or both...

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

Sat 29 Oct 2022 11:56:31 AM PDT

 x: 36209 y: 0 z: -36207        u:  2  v: 72416 
 x: 18106 y: 0 z: -18102        u:  4  v: 36208 
 x: 9056 y: 0 z: -9048        u:  8  v: 18104 
 x: 4534 y: 0 z: -4518        u:  16  v: 9052 
 x: 2279 y: 0 z: -2247        u:  32  v: 4526 
 x: 1199 y: 0 z: -1137        u:  62  v: 2336 
 x: 646 y: 0 z: -522        u:  124  v: 1168 
 x: 569 y: 0 z: -423        u:  146  v: 992 
 x: 416 y: 0 z: -168        u:  248  v: 584 
 x: 394 y: 0 z: -102        u:  292  v: 496 
 x: 394 y: 0 z: 102        u:  496  v: 292 
 x: 416 y: 0 z: 168        u:  584  v: 248 
 x: 569 y: 0 z: 423        u:  992  v: 146 
 x: 646 y: 0 z: 522        u:  1168  v: 124 
 x: 1199 y: 0 z: 1137        u:  2336  v: 62 
 x: 2279 y: 0 z: 2247        u:  4526  v: 32 
 x: 4534 y: 0 z: 4518        u:  9052  v: 16 
 x: 9056 y: 0 z: 9048        u:  18104  v: 8 
 x: 18106 y: 0 z: 18102        u:  36208  v: 4 
 x: 36209 y: 0 z: 36207        u:  72416  v: 2 

 x: 72415 y: 1 z: -72414        u:  1  v: 144829 
 x: 72415 y: 1 z: 72414        u:  144829  v: 1 

 x: 36206 y: 2 z: -36204        u:  2  v: 72410 
 x: 7246 y: 2 z: -7236        u:  10  v: 14482 
 x: 2798 y: 2 z: -2772        u:  26  v: 5570 
 x: 622 y: 2 z: -492        u:  130  v: 1114 
 x: 622 y: 2 z: 492        u:  1114  v: 130 
 x: 2798 y: 2 z: 2772        u:  5570  v: 26 
 x: 7246 y: 2 z: 7236        u:  14482  v: 10 
 x: 36206 y: 2 z: 36204        u:  72410  v: 2 

 x: 72403 y: 3 z: -72402        u:  1  v: 144805 
 x: 14483 y: 3 z: -14478        u:  5  v: 28961 
 x: 14483 y: 3 z: 14478        u:  28961  v: 5 
 x: 72403 y: 3 z: 72402        u:  144805  v: 1 

 x: 36197 y: 4 z: -36195        u:  2  v: 72392 
 x: 18100 y: 4 z: -18096        u:  4  v: 36196 
 x: 9053 y: 4 z: -9045        u:  8  v: 18098 
 x: 9053 y: 4 z: 9045        u:  18098  v: 8 
 x: 18100 y: 4 z: 18096        u:  36196  v: 4 
 x: 36197 y: 4 z: 36195        u:  72392  v: 2 

 x: 72379 y: 5 z: -72378        u:  1  v: 144757 
 x: 72379 y: 5 z: 72378        u:  144757  v: 1 

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

middle of the program

for(y = 0;  y <= 5; ++y ){
cout << endl;
           set<mpz_class>  div = mp_Divisors(  144832 - 3 * y * y); 
   set<mpz_class>::iterator iter;
   for(iter = div.begin(); iter != div.end(); ++iter)
   {
      u = *iter;
      v = (  144832 - 3 * y * y)  / u;
      if ( (u-v) % 2 == 0    )
      {
         x = (u+v)/ 2;
         z = (u-v)/ 2;
         cout << " x: " << x << " y: "  << y << " z: "  << z << "        u:  "  << u  << "  v: " << v <<" "  << endl;
       }  // even

   }  // for iter


}  // for y

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$