Investigating the rational roots of $f(x,y,z)=5x^2+7y^2-13z^2$.

Question

Question. Show that the quadratic form $f(x,y,z):=5x^2+7y^2-13z^2$ has a non-trivial rational root.

I am aware that to approach this problem, we need to use Hasse-Minkowski and investigate the behaviour of $f$ over $\Bbb Q_p^3$ and $\Bbb R^3$ before concluding over $\Bbb Q^3$. I have since found this exact example in this paper on page 20 and have a query pertaining to this.

• Why in the first case has the author looked at primes that don't divide $2,5,7,13$ specifically? It is probably something to do with the coefficients, however I notice that $2$ doesn't appear in the coefficients of $f$.

Thanks in advance.

Answer 1

The best treatment of this is Cassels, Rational Quadratic Forms. And 2 does divide the discriminant. Note that, for integer solutions, either $y$ or $z$ must be even. We get infinitely many (possibly all, requires more checking) primitive integer solutions by four recipes: $$ \color{blue}{ x = | \; 5 u^2 + 18uv - 2 v^2 \; | \; , \; \; y = | \; 7 u^2 -6uv - 8 v^2 \; | \; , \; \; z = \; 6 u^2 + 2uv + 6 v^2 \; \; . \; \;} $$

$$ \color{red}{ x = | \; 6 u^2 + 10uv - 11 v^2 \; | \; , \; \; y = | \; -2 u^2 +14uv + 8 v^2 \; | \; , \; \; z = \; 4 u^2 + 2uv + 9 v^2 \; \; . \; \;} $$

$$ \color{magenta}{ x = | \; 3 u^2 -4 uv - 29 v^2 \; | \; , \; \; y = | \; u^2 +16uv - v^2 \; | \; , \; \; z = \; 2 u^2 + 2uv + 18 v^2 \; \; . \; \;} $$

$$ \color{green}{ x = | \; u^2 - 18uv - 10 v^2 \; | \; , \; \; y = | \; 4 u^2 + 6uv -14 v^2 \; | \; , \; \; z = \; 3 u^2 + 2uv + 12 v^2 \; \; . \; \;} $$

Let's see, we take $u,v$ relatively prime integers in the above. It is still possible for $\gcd(x,y,z)$ to be larger than $1$ for some relationships between $u,v,$ if I have really gotten a complete list then the annoying imprimitive solutions can just be discarded. Indeed, in the first recipe we also need $u$ odd, in the second recipe we need $v$ odd. Recall similar such rules when parametrizing Pythagorean triples as $(m^2 - n^2, 2mn, m^2 + n^2)$ with coprime $m,n$ AND $m+n$ odd. I labelled the four recipes as A,B,C,D. Compared with a brute force search.

   2    x: 3    y: 1    z: 2       brute force   
   2    x: 3    y: 1    z: 2 C       u: 1    v: 0
   3    x: 1    y: 4    z: 3       brute force   
   3    x: 1    y: 4    z: 3 D       u: 1    v: 0
   6    x: 5    y: 7    z: 6 A       u: 1    v: 0
   6    x: 5    y: 7    z: 6       brute force   
   9    x: 11    y: 8    z: 9       brute force   
   9    x: 11    y: 8    z: 9 B       u: 0    v: 1
   9    x: 11    y: 8    z: 9 B       u: 0    v: -1
  11    x: 15    y: 8    z: 11       brute force   
  11    x: 15    y: 8    z: 11 B       u: 1    v: -1
  13    x: 9    y: 16    z: 13       brute force   
  13    x: 9    y: 16    z: 13 D       u: 1    v: -1
  17    x: 27    y: 4    z: 17       brute force   
  17    x: 27    y: 4    z: 17 D       u: 1    v: 1
  18    x: 29    y: 1    z: 18       brute force   
  18    x: 29    y: 1    z: 18 C       u: 0    v: 1
  18    x: 29    y: 1    z: 18 C       u: 0    v: -1
  22    x: 9    y: 29    z: 22       brute force   
  22    x: 9    y: 29    z: 22 C       u: 2    v: -1
  29    x: 33    y: 28    z: 29       brute force   
  29    x: 33    y: 28    z: 29 B       u: 2    v: 1
  33    x: 53    y: 4    z: 33       brute force   
  33    x: 53    y: 4    z: 33 D       u: 3    v: -1
  34    x: 33    y: 37    z: 34 A       u: 1    v: 2
  34    x: 33    y: 37    z: 34       brute force   
  47    x: 3    y: 64    z: 47       brute force   
  47    x: 3    y: 64    z: 47 D       u: 1    v: -2
  51    x: 73    y: 32    z: 51       brute force   
  51    x: 73    y: 32    z: 51 B       u: 3    v: 1
  54    x: 11    y: 73    z: 54 A       u: 3    v: -1
  54    x: 11    y: 73    z: 54       brute force   
  54    x: 67    y: 47    z: 54 A       u: 1    v: -3
  54    x: 67    y: 47    z: 54       brute force   
  58    x: 3    y: 79    z: 58       brute force   
  58    x: 3    y: 79    z: 58 C       u: 4    v: 1
  66    x: 41    y: 83    z: 66 A       u: 1    v: 3
  66    x: 41    y: 83    z: 66       brute force   
  66    x: 71    y: 67    z: 66 A       u: 3    v: -2
  66    x: 71    y: 67    z: 66       brute force   
  66    x: 97    y: 37    z: 66 A       u: 3    v: 1
  66    x: 97    y: 37    z: 66       brute force   
  78    x: 103    y: 61    z: 78       brute force   
  78    x: 103    y: 61    z: 78 C       u: 6    v: -1
  78    x: 121    y: 29    z: 78       brute force   
  78    x: 121    y: 29    z: 78 C       u: 1    v: 2
  79    x: 123    y: 28    z: 79       brute force   
  79    x: 123    y: 28    z: 79 B       u: 1    v: -3
  81    x: 125    y: 32    z: 81       brute force   
  81    x: 125    y: 32    z: 81 B       u: 4    v: 1
  87    x: 139    y: 16    z: 87       brute force   
  87    x: 139    y: 16    z: 87 D       u: 3    v: 2
  94    x: 99    y: 97    z: 94 A       u: 1    v: -4
  94    x: 99    y: 97    z: 94       brute force   
  97    x: 75    y: 116    z: 97       brute force   
  97    x: 75    y: 116    z: 97 D       u: 5    v: 1
  99    x: 89    y: 112    z: 99       brute force   
  99    x: 89    y: 112    z: 99 B       u: 5    v: -1
 102    x: 113    y: 101    z: 102       brute force   
 102    x: 113    y: 101    z: 102 C       u: 3    v: 2
 102    x: 1    y: 139    z: 102       brute force   
 102    x: 1    y: 139    z: 102 C       u: 5    v: -2
 102    x: 55    y: 131    z: 102       brute force   
 102    x: 55    y: 131    z: 102 C       u: 6    v: 1
 103    x: 165    y: 16    z: 103       brute force   
 103    x: 165    y: 16    z: 103 D       u: 5    v: -2 

---

   int x,y,z;


for(int u = 0; u <= 100; ++u){
for(int v = -100; v <= 100; ++v){




  x = abs( 5*u*u + 18 *u*v - 2*v*v  );
  y = abs( 7*u*u -6 *u*v - 8*v*v  );
  z =  6*u*u + 2 *u*v +6*v*v  ;

  if(coprime3(x,y,z) && z <= 1000 ) cout << setw(8) << z << "    x: " << x  << "    y: " << y  << "    z: " << z  << " A   "  << "    u: " << u  << "    v: " << v << endl; 

 
  x = abs( 6*u*u + 10 *u*v - 11*v*v  );
  y = abs( -2*u*u + 14 *u*v + 8*v*v  );
  z =  4*u*u + 2 *u*v +9*v*v  ;

  if(coprime3(x,y,z) && z <= 1000 ) cout << setw(8) << z << "    x: " << x  << "    y: " << y  << "    z: " << z  << " B   "  << "    u: " << u  << "    v: " << v << endl; 



  x = abs( 3*u*u -4 *u*v - 29*v*v  );
  y = abs( u*u + 16 *u*v - v*v  );
  z =  2*u*u + 2 *u*v +18*v*v  ;

  if(coprime3(x,y,z) && z <= 1000 ) cout << setw(8) << z << "    x: " << x  << "    y: " << y  << "    z: " << z  << " C   "  << "    u: " << u  << "    v: " << v << endl; 

  x = abs( u*u -18 *u*v - 10*v*v  );
  y = abs( 4 * u*u + 6 *u*v - 14 * v*v  );
  z =  3*u*u + 2 *u*v +12*v*v  ;

  if(coprime3(x,y,z) && z <= 1000 ) cout << setw(8) << z << "    x: " << x  << "    y: " << y  << "    z: " << z  << " D   "  << "    u: " << u  << "    v: " << v << endl;





}}


  for(z = 1; z <= 1000; ++z){
  for( x = 0; 5 * x*x <= 13 * z * z; ++x){
  for(y = 0; 5*x*x + 7*y*y <= 13*z*z; ++y){
   if ( 5*x*x + 7*y*y == 13*z*z  && coprime3(x,y,z) ) cout << setw(8) << z << "    x: " << x  << "    y: " << y  << "    z: " << z  << "       brute force   "  << endl; 

  }}}

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