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I was looking for integer solutions to this equation: $$a^2+b^2+c^2=6d^2$$ And found a parametric solution. Given u, v, w : $$a=(u-v)^2-2(u+w)^2+3w^2$$ $$b=(u-v)^2-2(v+w)^2+3w^2$$ $$c=(u-v)^2+(u+v-w)^2-3w^2$$ $$d=u^2+v^2+w^2$$

When will a parametric solution generate all possible solutions?

a^2 + b^2 + c^2 /. {a -> (u - v)^2 - 2 (u + w)^2 + 3 w^2,
b -> (u - v)^2 - 2 (v + w)^2 + 3 w^2,
c -> (u - v)^2 + (u + v - w)^2 - 3 w^2} // Factor

Mathematica Code

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The quick way is to take quaternions with integer coefficients, say $$ q = w + x i + y j + z k, $$ with $$ i^2 = j^2 = k^2 = -1, $$ then $$ ij = k, jk = i, k i = j $$ in cyclic order, then $$ ji = -k, kj = -i, ik = -j. $$ Let's see, multiplication is associative. define $$ \bar q = w - xi - yj - zk. $$ Oh, the norm of $q$ is $$ q \bar q = w^2 + x^2 + y^2 + z^2 $$ Now, take, as a "vector" $$ u = 2 i + j + k $$ of norm $6.$ There should be 24 possible vectors of norm $6.$

Finally, let $$ d = q \bar q \; . \; $$ The cute part is that $$ v = \bar q u q = ai + bj + c k $$ is another vector, no real part, and $$ a^2 + b^2 + c^2 = 6 d^2 $$ This way, we can deal with $d \equiv 7 \pmod 8,$ which cannot be written as the sum of three integer squares.

See what you can do with it. Note that, when $$ \gcd(a,b,c,d) = 1, $$ first $d$ is odd, then $a,b,c$ cannot be divisible by $3.$

$$\color{magenta}{ a = ( w+x+z)^2 + ( -w + x + y)^2 - 3 y^2 - 3 z^2 } $$

$$\color{magenta}{ b = ( w + x - 2 z)^2 - 2 ( x-y-z)^2 + 3 ( y+z)^2 - 6 z^2 } $$

$$\color{magenta}{ c = ( w - x + 2 y)^2 - 2 ( x-y-z)^2 - 3 ( y-z)^2 + 6 z^2 } $$

$$\color{magenta}{ d = w^2 + x^2 + y^2 + z^2 } $$

More later....1:45 pm my time.

Programmed it; very repetitive, as usual. The first thing I did was do a raw search, $a^2 + b^2 + c^2 = 6 d^2,$ with $\gcd(a,b,c) = 1,$ finally $a \geq b \geq c \geq 0.$ In that list, there are just three ways to get $d=7,$ just six ways to get $d=15.$

a = 2*x*x + (2*y + 2*z)*x + (2*w*w + (-2*y + 2*z)*w + (-2*y*y - 2*z*z)) ;

b = -x*x + (2*w + 4*y)*x + (w*w - 4*z*w + (y*y + 2*z*y - z*z)) ;

c = -x*x + (-2*w + 4*z)*x + (w*w + 4*y*w + (-y*y + 2*z*y + z*z)) ;

d = x*x + (w*w + (y*y + z*z)) ;

      1 a,b,c      -2     -1      1    w,x,y,z        0      0      0      1
      1 a,b,c      -2     -1      1    w,x,y,z        0      0      0     -1
      1 a,b,c      -2      1     -1    w,x,y,z        0      0      1      0
      1 a,b,c      -2      1     -1    w,x,y,z        0      0     -1      0
      1 a,b,c       2     -1     -1    w,x,y,z        0      1      0      0
      1 a,b,c       2     -1     -1    w,x,y,z        0     -1      0      0
      1 a,b,c       2      1      1    w,x,y,z        1      0      0      0
      3 a,b,c      -2      1     -7    w,x,y,z        0      1      1     -1
      3 a,b,c      -2      1     -7    w,x,y,z        0     -1     -1      1
      3 a,b,c      -2     -1      7    w,x,y,z        1      0      1      1
      3 a,b,c       2      1      7    w,x,y,z        1     -1      0     -1
      3 a,b,c       2     -1     -7    w,x,y,z        1      1     -1      0
      3 a,b,c       2      5      5    w,x,y,z        0      1      1      1
      3 a,b,c       2      5      5    w,x,y,z        0     -1     -1     -1
      3 a,b,c       2     -5     -5    w,x,y,z        1      0     -1      1
      3 a,b,c      -2      5     -5    w,x,y,z        1      1      0     -1
      3 a,b,c      -2     -5      5    w,x,y,z        1     -1      1      0
      3 a,b,c      -2     -7      1    w,x,y,z        0      1     -1      1
      3 a,b,c      -2     -7      1    w,x,y,z        0     -1      1     -1
      3 a,b,c      -2      7     -1    w,x,y,z        1      0     -1     -1
      3 a,b,c       2     -7     -1    w,x,y,z        1     -1      0      1
      3 a,b,c       2      7      1    w,x,y,z        1      1      1      0
      5 a,b,c     -10      1      7    w,x,y,z        0      0      1      2
      5 a,b,c     -10      1      7    w,x,y,z        0      0     -1     -2
      5 a,b,c     -10     -1     -7    w,x,y,z        0      0      2     -1
      5 a,b,c     -10     -1     -7    w,x,y,z        0      0     -2      1
      5 a,b,c      10      1     -7    w,x,y,z        1      2      0      0
      5 a,b,c      10     -1      7    w,x,y,z        2     -1      0      0
      5 a,b,c     -10     -7     -1    w,x,y,z        0      0      1     -2
      5 a,b,c     -10     -7     -1    w,x,y,z        0      0     -1      2
      5 a,b,c     -10      7      1    w,x,y,z        0      0      2      1
      5 a,b,c     -10      7      1    w,x,y,z        0      0     -2     -1
      5 a,b,c      10     -7      1    w,x,y,z        1     -2      0      0
      5 a,b,c      10      7     -1    w,x,y,z        2      1      0      0
      5 a,b,c      -2     11     -5    w,x,y,z        0      1      2      0
      5 a,b,c      -2     11     -5    w,x,y,z        0     -1     -2      0
      5 a,b,c       2    -11     -5    w,x,y,z        0      2     -1      0
      5 a,b,c       2    -11     -5    w,x,y,z        0     -2      1      0
      5 a,b,c      -2    -11      5    w,x,y,z        1      0      0      2
      5 a,b,c       2     11      5    w,x,y,z        2      0      0     -1
      5 a,b,c      -2     -5     11    w,x,y,z        0      1      0      2
      5 a,b,c      -2     -5     11    w,x,y,z        0     -1      0     -2
      5 a,b,c       2     -5    -11    w,x,y,z        0      2      0     -1
      5 a,b,c       2     -5    -11    w,x,y,z        0     -2      0      1
      5 a,b,c      -2      5    -11    w,x,y,z        1      0     -2      0
      5 a,b,c       2      5     11    w,x,y,z        2      0      1      0
      7 a,b,c     -10    -13      5    w,x,y,z        1     -1      1      2
      7 a,b,c     -10     13     -5    w,x,y,z        1      1      2     -1
      7 a,b,c      10    -13     -5    w,x,y,z        1      2     -1      1
      7 a,b,c      10     13      5    w,x,y,z        2     -1     -1     -1
      7 a,b,c     -10     -5     13    w,x,y,z        1     -1      1     -2
      7 a,b,c     -10      5    -13    w,x,y,z        1      1     -2     -1
      7 a,b,c      10     -5    -13    w,x,y,z        1     -2     -1      1
      7 a,b,c      10      5     13    w,x,y,z        2      1      1      1
      7 a,b,c      -2     11     13    w,x,y,z        1     -1     -1     -2
      7 a,b,c      -2    -11    -13    w,x,y,z        1      1     -2      1
      7 a,b,c       2     11    -13    w,x,y,z        1      2      1     -1
      7 a,b,c       2    -11     13    w,x,y,z        2     -1      1      1
      7 a,b,c       2     -1     17    w,x,y,z        1      1      1      2
      7 a,b,c       2      1    -17    w,x,y,z        1     -1     -2      1
      7 a,b,c      -2     -1    -17    w,x,y,z        1      2     -1     -1
      7 a,b,c      -2      1     17    w,x,y,z        2     -1      1     -1
      7 a,b,c      -2    -13    -11    w,x,y,z        1     -1     -1      2
      7 a,b,c      -2     13     11    w,x,y,z        1      1      2      1
      7 a,b,c       2    -13     11    w,x,y,z        1     -2      1     -1
      7 a,b,c       2     13    -11    w,x,y,z        2      1     -1     -1
      7 a,b,c       2    -17      1    w,x,y,z        1      1     -1      2
      7 a,b,c       2     17     -1    w,x,y,z        1     -1     -2     -1
      7 a,b,c      -2    -17     -1    w,x,y,z        1     -2      1      1
      7 a,b,c      -2     17      1    w,x,y,z        2      1      1     -1
      9 a,b,c      10     19      5    w,x,y,z        0      2      2      1
      9 a,b,c      10     19      5    w,x,y,z        0     -2     -2     -1
      9 a,b,c     -10    -19      5    w,x,y,z        1     -2      2      0
      9 a,b,c      10    -19     -5    w,x,y,z        2      0     -1      2
      9 a,b,c     -10     19     -5    w,x,y,z        2      1      0     -2
      9 a,b,c      10      5     19    w,x,y,z        0      2      1      2
      9 a,b,c      10      5     19    w,x,y,z        0     -2     -1     -2
      9 a,b,c     -10      5    -19    w,x,y,z        1      2      0     -2
      9 a,b,c      10     -5    -19    w,x,y,z        2      0     -2      1
      9 a,b,c     -10     -5     19    w,x,y,z        2     -1      2      0
      9 a,b,c     -14    -11    -13    w,x,y,z        0      2     -1     -2
      9 a,b,c     -14    -11    -13    w,x,y,z        0     -2      1      2
      9 a,b,c      14    -11     13    w,x,y,z        1      2      0      2
      9 a,b,c     -14     11     13    w,x,y,z        2      0      2     -1
      9 a,b,c      14     11    -13    w,x,y,z        2     -1     -2      0
      9 a,b,c     -14     -1    -17    w,x,y,z        0      1      2     -2
      9 a,b,c     -14     -1    -17    w,x,y,z        0     -1     -2      2
      9 a,b,c     -14      1     17    w,x,y,z        1      0      2      2
      9 a,b,c      14     -1     17    w,x,y,z        2     -2      0     -1
      9 a,b,c      14      1    -17    w,x,y,z        2      2     -1      0
      9 a,b,c     -14    -13    -11    w,x,y,z        0      2     -2     -1
      9 a,b,c     -14    -13    -11    w,x,y,z        0     -2      2      1
      9 a,b,c      14     13    -11    w,x,y,z        1     -2     -2      0
      9 a,b,c     -14     13     11    w,x,y,z        2      0      1     -2
      9 a,b,c      14    -13     11    w,x,y,z        2      1      0      2
      9 a,b,c     -14    -17     -1    w,x,y,z        0      1     -2      2
      9 a,b,c     -14    -17     -1    w,x,y,z        0     -1      2     -2
      9 a,b,c     -14     17      1    w,x,y,z        1      0     -2     -2
      9 a,b,c      14    -17      1    w,x,y,z        2     -2      0      1
      9 a,b,c      14     17     -1    w,x,y,z        2      2      1      0
      9 a,b,c       2     11    -19    w,x,y,z        0      2      2     -1
      9 a,b,c       2     11    -19    w,x,y,z        0     -2     -2      1
      9 a,b,c      -2    -11    -19    w,x,y,z        1      2     -2      0
      9 a,b,c       2    -11     19    w,x,y,z        2      0      1      2
      9 a,b,c      -2     11     19    w,x,y,z        2     -1      0     -2
      9 a,b,c       2    -19     11    w,x,y,z        0      2     -1      2
      9 a,b,c       2    -19     11    w,x,y,z        0     -2      1     -2
      9 a,b,c      -2    -19    -11    w,x,y,z        1     -2      0      2
      9 a,b,c       2     19    -11    w,x,y,z        2      0     -2     -1
      9 a,b,c      -2     19     11    w,x,y,z        2      1      2      0
      9 a,b,c     -22     -1     -1    w,x,y,z        0      1     -2     -2
      9 a,b,c     -22     -1     -1    w,x,y,z        0     -1      2      2
      9 a,b,c     -22      1      1    w,x,y,z        1      0      2     -2
      9 a,b,c      22     -1      1    w,x,y,z        2      2      0      1
      9 a,b,c      22      1     -1    w,x,y,z        2     -2     -1      0
     11 a,b,c     -10      1     25    w,x,y,z        0      1      1      3
     11 a,b,c     -10      1     25    w,x,y,z        0     -1     -1     -3
     11 a,b,c     -10     -1    -25    w,x,y,z        1      0     -3      1
     11 a,b,c      10      1    -25    w,x,y,z        1      3      0     -1
     11 a,b,c      10     -1     25    w,x,y,z        3     -1      1      0
     11 a,b,c     -10     25      1    w,x,y,z        0      1      3      1
     11 a,b,c     -10     25      1    w,x,y,z        0     -1     -3     -1
     11 a,b,c     -10    -25     -1    w,x,y,z        1      0     -1      3
     11 a,b,c      10    -25      1    w,x,y,z        1     -3      1      0
     11 a,b,c      10     25     -1    w,x,y,z        3      1      0     -1
     11 a,b,c      14      1    -23    w,x,y,z        0      3      1     -1
     11 a,b,c      14      1    -23    w,x,y,z        0     -3     -1      1
     11 a,b,c     -14      1     23    w,x,y,z        1     -1      0     -3
     11 a,b,c     -14     -1    -23    w,x,y,z        1      1     -3      0
     11 a,b,c      14     -1     23    w,x,y,z        3      0      1      1
     11 a,b,c     -14     13    -19    w,x,y,z        0      1      3     -1
     11 a,b,c     -14     13    -19    w,x,y,z        0     -1     -3      1
     11 a,b,c     -14    -13     19    w,x,y,z        1      0      1      3
     11 a,b,c      14    -13    -19    w,x,y,z        1      3     -1      0
     11 a,b,c      14     13     19    w,x,y,z        3     -1      0     -1
     11 a,b,c     -14    -19     13    w,x,y,z        0      1     -1      3
     11 a,b,c     -14    -19     13    w,x,y,z        0     -1      1     -3
     11 a,b,c     -14     19    -13    w,x,y,z        1      0     -3     -1
     11 a,b,c      14    -19    -13    w,x,y,z        1     -3      0      1
     11 a,b,c      14     19     13    w,x,y,z        3      1      1      0
     11 a,b,c      14    -23      1    w,x,y,z        0      3     -1      1
     11 a,b,c      14    -23      1    w,x,y,z        0     -3      1     -1
     11 a,b,c     -14    -23     -1    w,x,y,z        1     -1      0      3
     11 a,b,c     -14     23      1    w,x,y,z        1      1      3      0
     11 a,b,c      14     23     -1    w,x,y,z        3      0     -1     -1
     11 a,b,c       2    -19    -19    w,x,y,z        0      3     -1     -1
     11 a,b,c       2    -19    -19    w,x,y,z        0     -3      1      1
     11 a,b,c      -2    -19     19    w,x,y,z        1      1      0      3
     11 a,b,c      -2     19    -19    w,x,y,z        1     -1     -3      0
     11 a,b,c       2     19     19    w,x,y,z        3      0      1     -1
     11 a,b,c     -26      1     -7    w,x,y,z        0      1     -3     -1
     11 a,b,c     -26      1     -7    w,x,y,z        0     -1      3      1
     11 a,b,c     -26     -1      7    w,x,y,z        1      0      1     -3
     11 a,b,c      26     -1     -7    w,x,y,z        1     -3     -1      0
     11 a,b,c      26      1      7    w,x,y,z        3      1      0      1
     11 a,b,c      26      5      5    w,x,y,z        0      3      1      1
     11 a,b,c      26      5      5    w,x,y,z        0     -3     -1     -1
     11 a,b,c     -26      5     -5    w,x,y,z        1      1      0     -3
     11 a,b,c     -26     -5      5    w,x,y,z        1     -1      3      0
     11 a,b,c      26     -5     -5    w,x,y,z        3      0     -1      1
     11 a,b,c     -26     -7      1    w,x,y,z        0      1     -1     -3
     11 a,b,c     -26     -7      1    w,x,y,z        0     -1      1      3
     11 a,b,c     -26      7     -1    w,x,y,z        1      0      3     -1
     11 a,b,c      26     -7     -1    w,x,y,z        1      3      0      1
     11 a,b,c      26      7      1    w,x,y,z        3     -1     -1      0
     13 a,b,c      10     17     25    w,x,y,z        1      2      2      2
     13 a,b,c      10    -17    -25    w,x,y,z        2     -1     -2      2
     13 a,b,c     -10     17    -25    w,x,y,z        2      2     -1     -2
     13 a,b,c     -10    -17     25    w,x,y,z        2     -2      2     -1
     13 a,b,c      10     25     17    w,x,y,z        1     -2     -2     -2
     13 a,b,c      10    -25    -17    w,x,y,z        2      1     -2      2
     13 a,b,c     -10     25    -17    w,x,y,z        2      2      1     -2
     13 a,b,c     -10    -25     17    w,x,y,z        2     -2      2      1
     13 a,b,c     -14     17    -23    w,x,y,z        1      2      2     -2
     13 a,b,c     -14    -17     23    w,x,y,z        2     -1      2      2
     13 a,b,c      14     17     23    w,x,y,z        2     -2     -1     -2
     13 a,b,c      14    -17    -23    w,x,y,z        2      2     -2      1
     13 a,b,c     -14    -23     17    w,x,y,z        1     -2      2     -2
     13 a,b,c     -14     23    -17    w,x,y,z        2      1     -2     -2
     13 a,b,c      14    -23    -17    w,x,y,z        2     -2     -1      2
     13 a,b,c      14     23     17    w,x,y,z        2      2      2      1
     13 a,b,c       2    -13     29    w,x,y,z        0      2      0      3
     13 a,b,c       2    -13     29    w,x,y,z        0     -2      0     -3
     13 a,b,c      -2    -13    -29    w,x,y,z        0      3      0     -2
     13 a,b,c      -2    -13    -29    w,x,y,z        0     -3      0      2
     13 a,b,c       2     13    -29    w,x,y,z        2      0     -3      0
     13 a,b,c      -2     13     29    w,x,y,z        3      0      2      0
     13 a,b,c     -22      1    -23    w,x,y,z        1      2     -2     -2
     13 a,b,c     -22     -1     23    w,x,y,z        2     -1      2     -2
     13 a,b,c      22      1     23    w,x,y,z        2      2      1      2
     13 a,b,c      22     -1    -23    w,x,y,z        2     -2     -2      1
     13 a,b,c     -22    -13    -19    w,x,y,z        0      2      0     -3
     13 a,b,c     -22    -13    -19    w,x,y,z        0     -2      0      3
     13 a,b,c      22    -13     19    w,x,y,z        0      3      0      2
     13 a,b,c      22    -13     19    w,x,y,z        0     -3      0     -2
     13 a,b,c     -22     13     19    w,x,y,z        2      0      3      0
     13 a,b,c      22     13    -19    w,x,y,z        3      0     -2      0
     13 a,b,c     -22    -19    -13    w,x,y,z        0      2     -3      0
     13 a,b,c     -22    -19    -13    w,x,y,z        0     -2      3      0
     13 a,b,c      22     19    -13    w,x,y,z        0      3      2      0
     13 a,b,c      22     19    -13    w,x,y,z        0     -3     -2      0
     13 a,b,c     -22     19     13    w,x,y,z        2      0      0     -3
     13 a,b,c      22    -19     13    w,x,y,z        3      0      0      2
     13 a,b,c     -22    -23      1    w,x,y,z        1     -2      2      2
     13 a,b,c     -22     23     -1    w,x,y,z        2      1      2     -2
     13 a,b,c      22    -23     -1    w,x,y,z        2      2     -1      2
     13 a,b,c      22     23      1    w,x,y,z        2     -2     -2     -1
     13 a,b,c       2     29    -13    w,x,y,z        0      2      3      0
     13 a,b,c       2     29    -13    w,x,y,z        0     -2     -3      0
     13 a,b,c      -2    -29    -13    w,x,y,z        0      3     -2      0
     13 a,b,c      -2    -29    -13    w,x,y,z        0     -3      2      0
     13 a,b,c       2    -29     13    w,x,y,z        2      0      0      3
     13 a,b,c      -2     29     13    w,x,y,z        3      0      0     -2
     13 a,b,c       2    -31     -7    w,x,y,z        1      2     -2      2
     13 a,b,c       2     31      7    w,x,y,z        2     -1     -2     -2
     13 a,b,c      -2    -31      7    w,x,y,z        2     -2      1      2
     13 a,b,c      -2     31     -7    w,x,y,z        2      2      2     -1
     13 a,b,c     -26    -17     -7    w,x,y,z        0      0      2     -3
     13 a,b,c     -26    -17     -7    w,x,y,z        0      0     -2      3
     13 a,b,c     -26     17      7    w,x,y,z        0      0      3      2
     13 a,b,c     -26     17      7    w,x,y,z        0      0     -3     -2
     13 a,b,c      26    -17      7    w,x,y,z        2     -3      0      0
     13 a,b,c      26     17     -7    w,x,y,z        3      2      0      0
     13 a,b,c     -26      7     17    w,x,y,z        0      0      2      3
     13 a,b,c     -26      7     17    w,x,y,z        0      0     -2     -3
     13 a,b,c     -26     -7    -17    w,x,y,z        0      0      3     -2
     13 a,b,c     -26     -7    -17    w,x,y,z        0      0     -3      2
     13 a,b,c      26      7    -17    w,x,y,z        2      3      0      0
     13 a,b,c      26     -7     17    w,x,y,z        3     -2      0      0
     13 a,b,c       2     -7    -31    w,x,y,z        1     -2     -2      2
     13 a,b,c       2      7     31    w,x,y,z        2      1      2      2
     13 a,b,c      -2     -7     31    w,x,y,z        2     -2      1     -2
     13 a,b,c      -2      7    -31    w,x,y,z        2      2     -2     -1
     15 a,b,c     -10    -17     31    w,x,y,z        1     -2      1     -3
     15 a,b,c      10    -17    -31    w,x,y,z        1     -3     -1      2
     15 a,b,c     -10     17    -31    w,x,y,z        2      1     -3     -1
     15 a,b,c      10     17     31    w,x,y,z        3      1      2      1
     15 a,b,c     -10     31    -17    w,x,y,z        1      2      3     -1
     15 a,b,c      10    -31    -17    w,x,y,z        1      3     -2      1
     15 a,b,c     -10    -31     17    w,x,y,z        2     -1      1      3
     15 a,b,c      10     31     17    w,x,y,z        3     -1     -1     -2
     15 a,b,c     -14    -23    -25    w,x,y,z        1     -1     -2      3
     15 a,b,c     -14     23     25    w,x,y,z        1      1      3      2
     15 a,b,c      14    -23     25    w,x,y,z        2     -3      1     -1
     15 a,b,c      14     23    -25    w,x,y,z        3      2     -1     -1
     15 a,b,c     -14     25     23    w,x,y,z        1     -1     -2     -3
     15 a,b,c     -14    -25    -23    w,x,y,z        1      1     -3      2
     15 a,b,c      14     25    -23    w,x,y,z        2      3      1     -1
     15 a,b,c      14    -25     23    w,x,y,z        3     -2      1      1
     15 a,b,c       2     11     35    w,x,y,z        1     -2     -1     -3
     15 a,b,c      -2     11    -35    w,x,y,z        1      3      1     -2
     15 a,b,c       2    -11    -35    w,x,y,z        2      1     -3      1
     15 a,b,c      -2    -11     35    w,x,y,z        3     -1      2      1
     15 a,b,c     -22    -29     -5    w,x,y,z        1     -2      1      3
     15 a,b,c      22    -29      5    w,x,y,z        1      3     -1      2
     15 a,b,c     -22     29      5    w,x,y,z        2      1      3     -1
     15 a,b,c      22     29     -5    w,x,y,z        3     -1     -2     -1
     15 a,b,c     -22     -5    -29    w,x,y,z        1      2     -3     -1
     15 a,b,c      22      5    -29    w,x,y,z        1     -3     -2      1
     15 a,b,c     -22      5     29    w,x,y,z        2     -1      1     -3
     15 a,b,c      22     -5     29    w,x,y,z        3      1      1      2
     15 a,b,c       2     35     11    w,x,y,z        1      2      3      1
     15 a,b,c      -2    -35     11    w,x,y,z        1     -3      2     -1
     15 a,b,c       2    -35    -11    w,x,y,z        2     -1     -1      3
     15 a,b,c      -2     35    -11    w,x,y,z        3      1     -1     -2
     15 a,b,c     -26    -25      7    w,x,y,z        1     -2      3     -1
     15 a,b,c      26     25      7    w,x,y,z        1      3      2      1
     15 a,b,c     -26     25     -7    w,x,y,z        2      1     -1     -3
     15 a,b,c      26    -25     -7    w,x,y,z        3     -1     -1      2
     15 a,b,c     -26      7    -25    w,x,y,z        1      2      1     -3
     15 a,b,c      26      7     25    w,x,y,z        1     -3     -1     -2
     15 a,b,c     -26     -7     25    w,x,y,z        2     -1      3      1
     15 a,b,c      26     -7    -25    w,x,y,z        3      1     -2      1
     15 a,b,c     -34     13     -5    w,x,y,z        1      1     -2     -3
     15 a,b,c     -34    -13      5    w,x,y,z        1     -1      3     -2
     15 a,b,c      34     13      5    w,x,y,z        2      3      1      1
     15 a,b,c      34    -13     -5    w,x,y,z        3     -2     -1      1
     15 a,b,c     -34      5    -13    w,x,y,z        1      1      2     -3
     15 a,b,c     -34     -5     13    w,x,y,z        1     -1      3      2
     15 a,b,c      34      5     13    w,x,y,z        2     -3     -1     -1
     15 a,b,c      34     -5    -13    w,x,y,z        3      2     -1      1  
$\endgroup$
  • $\begingroup$ Thank you. I just found out that it become the result I gave after the substitution {w->0,x->w',y->v',z->u'}. But how can I express your answers in a more elegant form? LDLT Decomposition? Lagrange's reduction? $\endgroup$ – Eufisky Dec 2 '19 at 4:21
  • $\begingroup$ Thanks! \begin{cases} a= 2 (\alpha +\gamma +\delta )^2-(\alpha -\beta +2 \delta )^2-3 (\gamma +\delta )^2+6 \delta ^2\\ b= (\alpha -\beta +2 \delta )^2-2 (\beta +\gamma -\delta )^2+3 (\gamma -\delta )^2-6 \delta ^2\\ c= (\alpha +\beta -\gamma )^2+(\alpha -\beta -\delta )^2-3 \gamma ^2-3 \delta ^2\\ d=\alpha^2 +\beta^2 +\gamma^2 + \delta^2 \end{cases} $\endgroup$ – Eufisky Dec 2 '19 at 6:55
  • 1
    $\begingroup$ @Eufisky, good. Your $c$ is the one that is always even, then. Note that any primitive solution has an even and two odds out of a,b,c, because d must be odd, so 6 d^2 == 6 mod 8. Oh, and the even must be 2 mod 4, as its square needs to be 4 mod 8. Also, please note that $t^2 + u^2 + v^2 = 7 w^2$ is impossible, the only integer solution is $(0,0,0,0).$ About Lagrange's reduction, years ago someone showed me a reverse process, see math.stackexchange.com/questions/1388421/… $\endgroup$ – Will Jagy Dec 2 '19 at 17:51
1
$\begingroup$

I wrote a more careful program. It reports just one occurrence of each quadruple by the wxyz formulas i the first answer. I combined that with a raw search, so the computer would scream at me if any quadruple were missed. Success all around...

$$\color{magenta}{ a = ( w+x+z)^2 + ( -w + x + y)^2 - 3 y^2 - 3 z^2 } $$ $$\color{magenta}{ b = ( w + x - 2 z)^2 - 2 ( x-y-z)^2 + 3 ( y+z)^2 - 6 z^2 } $$ $$\color{magenta}{ c = ( w - x + 2 y)^2 - 2 ( x-y-z)^2 - 3 ( y-z)^2 + 6 z^2 } $$ $$\color{magenta}{ d = w^2 + x^2 + y^2 + z^2 } $$

  1 a,b,c       2      1      1
  1 a,b,c       2     -1     -1    w,x,y,z        0     -1      0      0

  3 a,b,c      -2      1     -7    w,x,y,z        0     -1     -1      1
  3 a,b,c       2      5      5    w,x,y,z        0     -1     -1     -1
  3 a,b,c       5      5      2
  3 a,b,c       7      2      1

  5 a,b,c      10      7      1
  5 a,b,c     -10      7      1    w,x,y,z        0      0     -2     -1
  5 a,b,c      11      5      2
  5 a,b,c       2     -5    -11    w,x,y,z        0     -2      0      1

  7 a,b,c      10     -5    -13    w,x,y,z        1     -2     -1      1
  7 a,b,c      13     10      5
  7 a,b,c      13     11      2
  7 a,b,c      17      2      1
  7 a,b,c       2    -13     11    w,x,y,z        1     -2      1     -1
  7 a,b,c      -2    -17     -1    w,x,y,z        1     -2      1      1

  9 a,b,c      10     19      5    w,x,y,z        0     -2     -2     -1
  9 a,b,c     -14    -11    -13    w,x,y,z        0     -2      1      2
  9 a,b,c     -14     -1    -17    w,x,y,z        0     -1     -2      2
  9 a,b,c      14     13     11
  9 a,b,c      17     14      1
  9 a,b,c      19     10      5
  9 a,b,c      19     11      2
  9 a,b,c       2     11    -19    w,x,y,z        0     -2     -2      1
  9 a,b,c      22      1      1
  9 a,b,c     -22     -1     -1    w,x,y,z        0     -1      2      2

 11 a,b,c     -10     25      1    w,x,y,z        0     -1     -3     -1
 11 a,b,c      14      1    -23    w,x,y,z        0     -3     -1      1
 11 a,b,c     -14     13    -19    w,x,y,z        0     -1     -3      1
 11 a,b,c      19     14     13
 11 a,b,c      19     19      2
 11 a,b,c       2    -19    -19    w,x,y,z        0     -3      1      1
 11 a,b,c      23     14      1
 11 a,b,c      25     10      1
 11 a,b,c      26      5      5
 11 a,b,c      26      5      5    w,x,y,z        0     -3     -1     -1
 11 a,b,c      26      7      1
 11 a,b,c     -26     -7      1    w,x,y,z        0     -1      1      3

 13 a,b,c      10     25     17    w,x,y,z        1     -2     -2     -2
 13 a,b,c     -14    -23     17    w,x,y,z        1     -2      2     -2
 13 a,b,c      -2    -13    -29    w,x,y,z        0     -3      0      2
 13 a,b,c      22     19     13
 13 a,b,c      22     19    -13    w,x,y,z        0     -3     -2      0
 13 a,b,c     -22    -23      1    w,x,y,z        1     -2      2      2
 13 a,b,c      23     17     14
 13 a,b,c      23     22      1
 13 a,b,c      25     17     10
 13 a,b,c      26     17      7
 13 a,b,c     -26     17      7    w,x,y,z        0      0     -3     -2
 13 a,b,c       2     -7    -31    w,x,y,z        1     -2     -2      2
 13 a,b,c      29     13      2
 13 a,b,c      31      7      2

 15 a,b,c      10    -17    -31    w,x,y,z        1     -3     -1      2
 15 a,b,c     -14     25     23    w,x,y,z        1     -1     -2     -3
 15 a,b,c      22      5    -29    w,x,y,z        1     -3     -2      1
 15 a,b,c      -2    -35     11    w,x,y,z        1     -3      2     -1
 15 a,b,c      25     23     14
 15 a,b,c      26     25      7
 15 a,b,c      26      7     25    w,x,y,z        1     -3     -1     -2
 15 a,b,c      29     22      5
 15 a,b,c      31     17     10
 15 a,b,c      34     13      5
 15 a,b,c     -34    -13      5    w,x,y,z        1     -1      3     -2
 15 a,b,c      35     11      2

 17 a,b,c     -14     13    -37    w,x,y,z        0     -2     -3      2
 17 a,b,c      22    -17    -31    w,x,y,z        0     -4      0      1
 17 a,b,c     -22    -25    -25    w,x,y,z        0     -3      2      2
 17 a,b,c     -22     -5    -35    w,x,y,z        0     -2     -2      3
 17 a,b,c       2     37     19    w,x,y,z        0     -2     -3     -2
 17 a,b,c      25     25     22
 17 a,b,c      26     23     23
 17 a,b,c      26     23     23    w,x,y,z        0     -3     -2     -2
 17 a,b,c       2      7    -41    w,x,y,z        0     -3     -2      2
 17 a,b,c      31     22     17
 17 a,b,c      34     23      7
 17 a,b,c     -34     23     -7    w,x,y,z        0      0     -4     -1
 17 a,b,c      35     22      5
 17 a,b,c      37     14     13
 17 a,b,c      37     19      2
 17 a,b,c      38      1    -17    w,x,y,z        0     -4     -1      0
 17 a,b,c      38     13     11
 17 a,b,c     -38    -13    -11    w,x,y,z        0     -2      2      3
 17 a,b,c      38     17      1
 17 a,b,c      41      7      2

 19 a,b,c      10     29    -35    w,x,y,z        0     -3     -3      1
 19 a,b,c     -14    -11    -43    w,x,y,z        0     -3     -1      3
 19 a,b,c      14    -41    -17    w,x,y,z        1     -4      1      1
 19 a,b,c     -22     29     29    w,x,y,z        0     -1     -3     -3
 19 a,b,c      22     41      1    w,x,y,z        0     -3     -3     -1
 19 a,b,c     -26    -23    -31    w,x,y,z        0     -3      1      3
 19 a,b,c      26    -37     11    w,x,y,z        1     -4      1     -1
 19 a,b,c      29     29     22
 19 a,b,c      31     26     23
 19 a,b,c      34    -13    -29    w,x,y,z        1     -4     -1      1
 19 a,b,c      34     29     13
 19 a,b,c      34     31      7
 19 a,b,c     -34     -7    -31    w,x,y,z        0     -1     -3      3
 19 a,b,c      35     29     10
 19 a,b,c      37     26     11
 19 a,b,c      41     17     14
 19 a,b,c      41     22      1
 19 a,b,c      43     14     11
 19 a,b,c      46     -1      7    w,x,y,z        1     -4     -1     -1
 19 a,b,c      46      5      5
 19 a,b,c     -46      5      5    w,x,y,z        0     -1      3      3
 19 a,b,c      46      7      1

 21 a,b,c      -2    -31    -41    w,x,y,z        0     -4      1      2
 21 a,b,c      26    -17    -41    w,x,y,z        2     -3     -2      2
 21 a,b,c     -26     43     11    w,x,y,z        0     -1     -4     -2
 21 a,b,c     -34     11    -37    w,x,y,z        0     -1     -4      2
 21 a,b,c      34     31     23
 21 a,b,c      34     31     23    w,x,y,z        2     -3     -2     -2
 21 a,b,c      37     34     11
 21 a,b,c     -38    -19    -29    w,x,y,z        0     -2     -1      4
 21 a,b,c      38     29     19
 21 a,b,c      41     26     17
 21 a,b,c      41     31      2
 21 a,b,c      43     26     11
 21 a,b,c      46     19     13
 21 a,b,c     -46    -19    -13    w,x,y,z        0     -2      1      4
 21 a,b,c      46     23      1
 21 a,b,c      46     23      1    w,x,y,z        0     -4     -2     -1
 21 a,b,c      50     11      5
 21 a,b,c     -50     -5     11    w,x,y,z        0     -1      2      4

 23 a,b,c     -10    -43     35    w,x,y,z        1     -3      2     -3
 23 a,b,c      10      7    -55    w,x,y,z        1     -3     -3      2
 23 a,b,c     -14    -13    -53    w,x,y,z        1     -2     -3      3
 23 a,b,c      -2    -19    -53    w,x,y,z        1     -3     -2      3
 23 a,b,c      22     29     43    w,x,y,z        1     -3     -2     -3
 23 a,b,c     -22    -49     17    w,x,y,z        1     -3      3     -2
 23 a,b,c      -2     47     31    w,x,y,z        1     -2     -3     -3
 23 a,b,c      26     47     17    w,x,y,z        1     -3     -3     -2
 23 a,b,c     -34    -43    -13    w,x,y,z        1     -3      2      3
 23 a,b,c      38     37     19
 23 a,b,c     -38    -37     19    w,x,y,z        1     -2      3     -3
 23 a,b,c     -38    -41     -7    w,x,y,z        1     -3      3      2
 23 a,b,c      41     38      7
 23 a,b,c      43     29     22
 23 a,b,c      43     34     13
 23 a,b,c      43     35     10
 23 a,b,c      47     26     17
 23 a,b,c      47     31      2
 23 a,b,c      49     22     17
 23 a,b,c      50     25      7
 23 a,b,c     -50    -25      7    w,x,y,z        1     -2      3      3
 23 a,b,c      53     14     13
 23 a,b,c      53     19      2
 23 a,b,c      55     10      7 
$\endgroup$

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