Restrictions on the [hypothetical] solutions of a ternary quadratic form

Question

Let $r,s,t,x,y,z$ be integers, with $rst$ squarefree, such that $$rx^2+sy^2+tz^2=0.$$

I already know that by [one of] Legendre's famous theorem[s], $-rs$ must be a square modulo each prime divisor of $t$, and $-rt$ must be a square modulo each prime divisor of $s$, and $-st$ must be a square modulo each prime divisor of $r$.

Are there any similar tests that can be applied to $x,y,z$ (either in terms of themselves, or $r,s,t$, or all of the above) in order to restrict and/or eliminate possible solutions?

Answer 1

I made one up of the kind you wanted, $$ 3 x^2 = 5 y^2 + 7 z^2. $$ This has a solution because $12 = 5 + 7.$ I was quite surprised that it took four of the Fricke-Klein recipes to account for all of the first three dozen or so primitive integer solutions. I probably have them all, but that needs a proof. First I will put the solutions with $x$ up to some bound, then the section of the program that shows the four 3 by 3 matrices as coefficients of binary quadratic forms. Maybe tomorrow I will also typeset that.

Why not... the four coefficient matrices are

$$ \left( \begin{array}{rrr} 18 & 2 & 2 \\ 12 & 6 & -1 \\ -6 & 6 & 1 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 12 & -2 & 3 \\ 6 & 6 & -2 \\ -6 & 6 & 1 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 11 & 6 & 4 \\ 2 & 10 & 2 \\ -7 & -2 & 2 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 26 & 22 & 6 \\ 1 & 10 & 4 \\ 17 & 14 & 2 \end{array} \right) $$

Next morning. This made much more sense after I looked at the positive binary forms of discriminant $-140,$ including the two imprimitive $\langle 2,2,18 \rangle$ and $\langle 6,2,6 \rangle,$ primitive forms (out of six total) $\langle 3,2,12 \rangle$ and $\langle 4,2,9 \rangle.$ I could, with the same results, have taken the four matrices as

$$ \left( \begin{array}{rrr} 2 & 2 & 18 \\ -1 & -8 & 5 \\ 1 & -4 & -11 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 6 & 2 & 6 \\ 4 & 6 & -3 \\ 2 & -6 & -3 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 3 & 2 & 12 \\ -2 & -6 & 6 \\ 1 & -6 & -6 \end{array} \right) $$

$$ \left( \begin{array}{rrr} 4 & 2 & 9 \\ 2 & -6 & -6 \\ 2 & 6 & -3 \end{array} \right) $$

One thing you can do by hand, to understand this a little better, is take each of these four matrices, one at a time, call it $P,$ and calculate $P^T HP,$ where $H$ is the diagonal matrix $\operatorname{diag}(-3,5,7)$

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   x       y       z                     u  v     x form:    
   2      -1       1      gcd    1       0  1     18 2 2 
   2      -1       1      gcd    1       0 -1     18 2 2 
   3      -2       1      gcd    1       0  1     12 -2 3 
   3      -2       1      gcd    1       0 -1     12 -2 3 
  11       2      -7      gcd    1       1  0     11 6 4 
  13      10       1      gcd    1       1  1     12 -2 3 
  17      -2     -11      gcd    1       1 -1     12 -2 3 
  18       5     -11      gcd    1       1 -1     18 2 2 
  22      17       1      gcd    1       1  1     18 2 2 
  26       1      17      gcd    1       1  0     26 22 6 
  29     -10      17      gcd    1       1 -3     11 6 4 
  34      25      -7      gcd    1       1 -4     26 22 6 
  47      34     -11      gcd    1       2  1     12 -2 3 
  54     -41      -7      gcd    1       3 -5     26 22 6 
  58     -43     -11      gcd    1       1 -5     18 2 2 
  66     -47      17      gcd    1       3 -4     26 22 6 
  66      -5     -43      gcd    1       3 -7     26 22 6 
  78      17      49      gcd    1       1  5     18 2 2 
  78      59     -11      gcd    1       2  1     18 2 2 
  79     -34     -43      gcd    1       3 -2     11 6 4 
  81       2      53      gcd    1       1 -5     11 6 4 
  94      37      53      gcd    1       1  2     26 22 6 
  97     -74     -11      gcd    1       1 -5     12 -2 3 
  99      74      17      gcd    1       1  4     11 6 4 
 102     -37     -59      gcd    1       2 -5     18 2 2 
 102     -79       1      gcd    1       1 -7     18 2 2 
 103      34      61      gcd    1       2  5     12 -2 3 
 109     -82      17      gcd    1       3 -5     11 6 4 
 117      34     -71      gcd    1       3 -1     12 -2 3 
 121      50     -67      gcd    1       3  1     11 6 4 
 142      83      61      gcd    1       2  5     18 2 2 
 142     -85     -59      gcd    1       2 -7     18 2 2 
 143     -86     -59      gcd    1       2 -5     12 -2 3 
 146    -101     -43      gcd    1       5 -9     26 22 6 
 146      67      77      gcd    1       1  3     26 22 6 
 151      86     -67      gcd    1       3  2     11 6 4 
 153      94      61      gcd    1       3  5     12 -2 3 
 158      89     -71      gcd    1       3 -1     18 2 2 
 166     127      17      gcd    1       1 -7     26 22 6 
 166     -41    -103      gcd    1       5-11     26 22 6 
 167      10     109      gcd    1       2  7     12 -2 3 
 169     -94      77      gcd    1       3 -7     11 6 4 
 173    -134       1      gcd    1       1 -7     12 -2 3 
 174     109     -67      gcd    1       3-10     26 22 6 
 174     -17     113      gcd    1       3 -1     26 22 6 
 187     118     -71      gcd    1       4  1     12 -2 3 
 191     134      53      gcd    1       1  6     11 6 4 
 194       1    -127      gcd    1       5-12     26 22 6 
 198      83     109      gcd    1       2  7     18 2 2 
 206    -131      77      gcd    1       5 -6     26 22 6 
 211     -94     113      gcd    1       3 -8     11 6 4 
 213      82     121      gcd    1       3  7     12 -2 3 
 213     -86    -119      gcd    1       3 -5     12 -2 3 
 218     -67    -131      gcd    1       3 -7     18 2 2 
 219    -118    -103      gcd    1       5 -4     11 6 4 
 219      50     137      gcd    1       1 -8     11 6 4 
 221     -82    -127      gcd    1       5 -3     11 6 4 
 223    -158     -59      gcd    1       2 -7     12 -2 3 
 227     166      49      gcd    1       4  5     12 -2 3 
 234     163     -67      gcd    1       3-11     26 22 6 
 239    -178     -43      gcd    1       5 -6     11 6 4 
 242     173      61      gcd    1       3  5     18 2 2 
 249     170      77      gcd    1       1  7     11 6 4 
 249       2    -163      gcd    1       5 -1     11 6 4 
 261    -202      -7      gcd    1       5 -7     11 6 4 
 281      74     173      gcd    1       1 -9     11 6 4 
 282     167    -119      gcd    1       4 -1     18 2 2 
 282     -43     181      gcd    1       1 11     18 2 2 
 283     166     121      gcd    1       4  7     12 -2 3 
 286     109    -163      gcd    1       5-14     26 22 6 
 286     151     137      gcd    1       1  5     26 22 6 
 286    -185    -103      gcd    1       7-13     26 22 6 
 289     218     -43      gcd    1       3  5     11 6 4 
 293     178    -119      gcd    1       5  1     12 -2 3 
 297    -170    -131      gcd    1       3 -7     12 -2 3 
 298     215     -71      gcd    1       4  1     18 2 2 
 298      47    -191      gcd    1       4 -5     18 2 2 
 302     185     121      gcd    1       3  7     18 2 2 
 306      43     197      gcd    1       3  1     26 22 6 
 307     -74    -191      gcd    1       4 -5     12 -2 3 
 313     118    -179      gcd    1       5 -1     12 -2 3 
 314    -101    -187      gcd    1       7-15     26 22 6 
 314     235      53      gcd    1       1 -9     26 22 6 
 319     -82     197      gcd    1       3-10     11 6 4 
 334    -251      53      gcd    1       7-10     26 22 6 
 338    -211    -131      gcd    1       3-11     18 2 2 
 346     -47    -223      gcd    1       7-16     26 22 6 
 351     158    -187      gcd    1       5  2     11 6 4 
 353    -170     181      gcd    1       1 11     12 -2 3 
 358    -277     -11      gcd    1       2-13     18 2 2 
 358      59     229      gcd    1       2 11     18 2 2 
 367     -86     229      gcd    1       2 11     12 -2 3 
 374     205     173      gcd    1       1  6     26 22 6 
 374    -257     113      gcd    1       7 -9     26 22 6 
 374     -89     233      gcd    1       5 -3     26 22 6 
 377     262     109      gcd    1       5  7     12 -2 3 
 382     -79     241      gcd    1       1 13     18 2 2 
 389     254     137      gcd    1       1  9     11 6 4 
 397    -302      49      gcd    1       1-11     12 -2 3 
 401     218    -187      gcd    1       5  3     11 6 4 
 421     326      -7      gcd    1       3  7     11 6 4 
 422    -295    -119      gcd    1       3-13     18 2 2 
 423     250    -179      gcd    1       6  1     12 -2 3 
 429     134     257      gcd    1       1-11     11 6 4 
 429    -202    -223      gcd    1       7 -5     11 6 4 
 431    -250    -187      gcd    1       7 -6     11 6 4 
 442    -193    -239      gcd    1       4-11     18 2 2 
 442     269    -179      gcd    1       5 -1     18 2 2 
 442     311     121      gcd    1       4  7     18 2 2 
 446      85    -283      gcd    1       7-18     26 22 6 
 447     178    -251      gcd    1       6 -1     12 -2 3 
 447     346     -11      gcd    1       6  5     12 -2 3 
 449     -94    -283      gcd    1       7 -3     11 6 4 
 454     -59     293      gcd    1       5 -2     26 22 6 
 459    -334    -103      gcd    1       7 -8     11 6 4 
 467     118     289      gcd    1       4 11     12 -2 3 
 471     302     173      gcd    1       1 10     11 6 4 
 471     -34    -307      gcd    1       7 -2     11 6 4 
 474    -293    -187      gcd    1       9-17     26 22 6 
 474    -335    -127      gcd    1       9-16     26 22 6 
 478      41    -311      gcd    1       5 -7     18 2 2 
 486     373     -43      gcd    1       3-14     26 22 6 
 491    -262     233      gcd    1       5-12     11 6 4 
 493    -254     241      gcd    1       1 13     12 -2 3 
 499     386      17      gcd    1       3  8     11 6 4 

jagy@phobeusjunior:~$

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 if ( twogcd(u,v) == 1)
    {
      int x = 18 * u * u + 2 * u * v + 2 * v * v;
      int y = 12 * u * u + 6 * u * v - 1 * v * v;
      int z = -6 * u * u + 6 * u * v + 1 * v * v;
      int g = threegcd(x,y,z);
      if ( x < 500 && g == 1)      cout << setw(8) << x  << setw(8) << y  << setw(8) << z  << "      gcd " << setw(4) << g  << setw(8) << u  << setw(3) << v << "     18 2 2 " << endl;
       x = 12 * u * u - 2 * u * v + 3 * v * v;
       y = 6 * u * u + 6 * u * v - 2 * v * v;
       z = -6 * u * u + 6 * u * v + 1 * v * v;
       g = threegcd(x,y,z);
      if ( x < 500 && g == 1)      cout << setw(8) << x  << setw(8) << y  << setw(8) << z  << "      gcd " << setw(4) << g  << setw(8) << u  << setw(3) << v << "     12 -2 3 " << endl;
       x = 11 * u * u + 6 * u * v + 4 * v * v;
       y = 2 * u * u + 10 * u * v + 2 * v * v;
       z = -7 * u * u - 2 * u * v + 2 * v * v;
       g = threegcd(x,y,z);
      if ( x < 500 && g == 1)      cout << setw(8) << x  << setw(8) << y  << setw(8) << z  << "      gcd " << setw(4) << g  << setw(8) << u  << setw(3) << v << "     11 6 4 " << endl;
             x = 26 * u * u + 22 * u * v + 6 * v * v;
       y = 1 * u * u + 10 * u * v + 4 * v * v;
       z = 17 * u * u + 14 * u * v + 2 * v * v;
       g = threegcd(x,y,z);
      if ( x < 500 && g == 1)      cout << setw(8) << x  << setw(8) << y  << setw(8) << z  << "      gcd " << setw(4) << g  << setw(8) << u  << setw(3) << v << "     26 22 6 " << endl;
    }

====================================

Answer 2

Just in case: if there are any rational solutions, then any such can be multiplied by a common denominator to give an integer solution.

This bit goes back to Fricke and Klein (1897). All primitive integer solutions can be produced by a finite number of integer parametrizations of the same type as Pythagorean triples. The parametrizations are all of type $$ \left( A u^2 + B uv + C v^2, \; \; D u^2 + E uv + F v^2, \; \; G u^2 + H uv + I v^2 \right), $$ where we take only pairs with $\gcd(u,v).$ For each prime divisor of $$ \det \left( \begin{array}{rrr} A & B & C \\ D & E & F \\ G & H & I \end{array} \right), $$ there will be a common divisor of $ \left( A u^2 + B uv + C v^2, \; D u^2 + E uv + F v^2, \; G u^2 + H uv + I v^2 \right) $ for some linear relationship of $u,v \pmod p.$ It is then a question of whether that triple, divided through by the gcd, is represented by some other $(u,v)$ pair, or perhaps another 3 by 3 matrix is needed. Each such 3 by 3 matrix means a triple of binary quadratic forms. For Pythagorean triples, we usually take

$$ \left( \begin{array}{rrr} 1 & 0 & -1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{array} \right) $$ with determinant $4.$ The only requirement beyond $\gcd(u,v) = 1$ is that $u + v \neq 0 \pmod 2.$

That being said, any one such parametrizations gives all rational solutions, it is just a mess pulling out all the primitive integer ones.