Rational solutions of the system $ p^2-2 q^2 + r^2 =12, 2p^2 - 3 q^2 + s^2=42$?

Question

I am looking for a rational solution of the system $ p^2-2 q^2 + r^2 =12, 2p^2 - 3 q^2 + s^2=42$ for $p,q,r$ with $p\ne 0$.

Does it have any?

Answer 1

I found integer solutions to $$ p^2 - 2 q^2 + r^2 = 12 v^2 $$ $$ 2p^2 - 3 q^2 + s^2 = 42 w^2 $$ When $v=w$ this gives a rational solution to your original question.

Thu Dec 19 15:31:50 PST 2019

  p:  1   q:   1  r:   7  s:  13    12 * 2^2      42 * 2^2  
  p:  1   q:  11  r:  17  s:  23    12 * 2^2      42 * 2^2  
  p:  2   q:   2  r:   4  s:  26    12 * 1^2      42 * 4^2  
  p:  2   q:   2  r:  14  s:  26    12 * 4^2      42 * 4^2  
  p:  5   q:   1  r:   5  s:  11    12 * 2^2      42 * 2^2  
  p:  5   q:   1  r:   5  s:  25    12 * 2^2      42 * 4^2  
  p:  5   q:   1  r:  13  s:  11    12 * 4^2      42 * 2^2  
  p:  5   q:   1  r:  13  s:  25    12 * 4^2      42 * 4^2  
  p:  5   q:  13  r:  19  s:  25    12 * 2^2      42 * 2^2  
  p:  10  q:   2  r:   4  s:  22    12 * 3^2      42 * 4^2  
  p:  10  q:   2  r:  10  s:  22    12 * 4^2      42 * 4^2  
  p:  10  q:   2  r:  26  s:  22    12 * 8^2      42 * 4^2  
  p:  11  q:   5  r:  11  s:   1    12 * 4^2      42 * 2^2  
  p:  11  q:   5  r:  19  s:   1    12 * 6^2      42 * 2^2  
  p:  11  q:  11  r:  13  s:  17    12 * 2^2      42 * 2^2  
  p:  13  q:  13  r:  19  s:  29    12 * 4^2      42 * 4^2  
  p:  17  q:   5  r:  23  s:  13    12 * 8^2      42 * 4^2  
  p:  19  q:   5  r:  11  s:   5    12 * 6^2      42 * 4^2  
  p:  22  q:  10  r:   4  s:   2    12 * 5^2      42 * 4^2  
  p:  22  q:  10  r:  22  s:   2    12 * 8^2      42 * 4^2  
  p:  23  q:   5  r:  17  s:  23    12 * 8^2      42 * 6^2  
  p:  23  q:  13  r:   1  s:  11    12 * 4^2      42 * 4^2  
  p:  25  q:  11  r:   7  s:  25    12 * 6^2      42 * 6^2  
  p:  25  q:  17  r:   1  s:  17    12 * 2^2      42 * 4^2  
  p:  25  q:  19  r:  17  s:   1    12 * 4^2      42 * 2^2  
  p:  25  q:  19  r:  23  s:   1    12 * 6^2      42 * 2^2  
  p:  29  q:  25  r:  29  s:  19    12 * 6^2      42 * 2^2  


Thu Dec 19 15:31:52 PST 2019

.............................

Language is C++ with GMP, I use a number of commands that I defined elsewere, each a little program...

  mpz_class bound = 150 ;

  for(mpz_class p = 0; p <= bound; ++p){
  for(mpz_class q = 0; q <= bound; ++q){
  for(mpz_class r = 0; r <= bound; ++r){
  for(mpz_class s = 0; s <= bound; ++s){

    mpz_class pqr = p*p - 2*q*q + r*r ;
    mpz_class pqs = 2*p*p - 3*q*q + s*s;
   if ( pqr > 0 && pqs > 0 && pqr % 12 == 0 && pqs %42 == 0) {
      pqr /= 12;
      pqs /= 42;

     if (mp_SquareQ(pqr) && mp_SquareQ(pqs) )
     {
        pqr = mp_Sqrt(pqr);
              pqs = mp_Sqrt(pqs);
       cout << "  p:  " << p << "  q:  " << q  << "  r:  " << r  << "  s:  " << s <<   "    "  <<  12 << " * " << pqr << "^2  " <<   "    "  <<  42 << " * " << pqs << "^2  "<< endl;
     }

   }


  }}}}

Answer 2

$$p^2-2q^2+r^2 = 12$$ $$2p^2-3q^2+s^2 = 42$$ From first equation, we get the solution $$p = 4(-5+k^2-k)/((k-1)(k+1))$$ $$q = -2(7+k^2+2k)/((k-1)(k+1))$$ $$r = -2(1+k^2+8k)/((k-1)(k+1)).$$

k is arbitrary.
Substitute $p,q,r$ to second equation, we get $$s^2 = 2(-85+210k^2+8k+11k^4+56k^3)/((k-1)^2(k+1)^2).$$ This quartic equation can be transformed to the elliptic curve $$Y^2=X^3 -2628X + 114048.$$
This elliptic curve has rank 2 with generator $P1(X,Y)=(-48,360), P2(X,Y)=(-6,360)$.
Thus, this elliptic curve has infinitely many rational points.
$P1: (p,q,r,s)=(\frac{5711}{182}, \frac{4619}{182},\frac{3233}{182},\frac{407}{182})$
$P1+P2: (p,q,r,s)=(\frac{1}{2}, \frac{11}{2},\frac{17}{2},\frac{23}{2})$
$2P1: (p,q,r,s)=(\frac{1006778053009}{14789940440}, \frac{918038410369}{14789940440},\frac{821347615729}{14789940440},\frac{714399503711}{14789940440})$