Determine all integers $x,\ y,\ z$ that satisfy $x+y+z=(x-y)^{2}+(y-z)^{2}+(z-x)^{2}$

Solution 1:

Let $y=u+x$ and $z=v+x$, then $y-z = u-v$ and hence $$ \begin{align} (y-u) + y + (y-u+v) &= u^2 + (u-v)^2 + v^2\\ 3y &= 2u^2 + 2v^2 - 2uv +2u - v \end{align} $$ So we can almost freely choose $u,v$, but we need RHS to be divisible by $3$. By considering the equation modulo $3$, we see that solutions are $$ (u,v) \equiv (0,0),(0,2),(1,1),(1,2),(2,0),(2,1) \pmod 3 $$ For example, taking $(u,v)\equiv (0,0)\pmod 3$, this gives parametrizations: $$ \begin{align} u = 3m, v = 3n &\implies y= 2 m + 6 m^2 - n - 6 m n + 6 n^2\\ &\implies x = -m + 6 m^2 - n - 6 m n + 6 n^2\\ &\implies z = -m + 6 m^2 + 2n - 6 m n + 6 n^2 \end{align} $$ where we can freely choose $m,n\in\mathbb Z$. The other cases are the same.

Edit 1: One more case: let $(u,v)\equiv (2,1)\pmod 3$, so that $$ u = 3m+2,\quad v = 3n+1 $$ then $$ \begin{align} y &= 3 + 8 m + 6 m^2 - n - 6 m n + 6 n^2\\ x = y-u &= 1 + 5 m + 6 m^2 - n - 6 m n + 6 n^2\\ z =v+x &= 2 + 5 m + 6 m^2 + 2 n - 6 m n + 6 n^2 \end{align} $$

Solution 2:

The surface is a paraboloid of revolution around the line $x=y=z$

If we define $$ u = x+y+z, \; \; v = -x + y, \; \; w = -x -y + 2z, $$ then we demand $$ v \equiv w \pmod 2 \; , $$ take solutions of $$ 2u = 3 v^2 + w^2 \; , $$ and further demand $$ u + w \equiv 0 \pmod 3 \; , $$ we have a solution to the original problem. That is $$ x = \frac{2u-3v-w}{6} \; , \; \; y = \frac{2u+3v-w}{6} \; , \; \; z = \frac{u+w}{3} \; . $$ Negating $v$ just interchanges $x,y$ so we might as well take $v \geq 0.$ Other than that, I just took $v \equiv w \pmod 2,$ next $u = \frac{3v^2 + w^2}{2}.$ Sometimes this $u$ was not suitable $\pmod 3,$ so I printed out things only when $u + w \equiv 0 \pmod 3.$

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

 int bound = 10;

  for(int wabs = 0; wabs <= bound; ++wabs){

  int vstart = wabs % 2;

  for(int v = vstart; v <= bound; v += 2){
  for(int wsign = 1; wsign >= -1; wsign -= 2) {   
   int w = wsign * wabs;
   int u = ( 3 * v * v + w * w  ) / 2;

   if( abs(u + w ) % 3 == 0 )
   {
      int z = (u + w)/3;
      int y = ( 2 * u - 3 * v - w) / 6 ;
      int x = ( 2 * u + 3 * v - w) / 6 ;
   cout << setw(6) << x  << setw(6) << y  << setw(6) << z   << "               "  << setw(6) << u  << setw(6) << v  << setw(6) << w   << endl;
   } // if 3

  }}}

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

 x     y     z                    u     v     w
 0     0     0                    0     0     0
 0     0     0                    0     0     0
 3     1     2                    6     2     0
 3     1     2                    6     2     0
10     6     8                   24     4     0
10     6     8                   24     4     0
21    15    18                   54     6     0
21    15    18                   54     6     0
36    28    32                   96     8     0
36    28    32                   96     8     0
55    45    50                  150    10     0
55    45    50                  150    10     0
 1     0     1                    2     1     1
 6     3     5                   14     3     1
15    10    13                   38     5     1
28    21    25                   74     7     1
45    36    41                  122     9     1
 1     1     0                    2     0    -2
 4     2     2                    8     2    -2
11     7     8                   26     4    -2
22    16    18                   56     6    -2
37    29    32                   98     8    -2
56    46    50                  152    10    -2
 2     1     3                    6     1     3
 3     2     1                    6     1    -3
 7     4     7                   18     3     3
 8     5     5                   18     3    -3
16    11    15                   42     5     3
17    12    13                   42     5    -3
29    22    27                   78     7     3
30    23    25                   78     7    -3
46    37    43                  126     9     3

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

sort by x,y,z

 x     y     z                    u     v     w
 1     0     1                    2     1     1
 1     1     0                    2     0    -2
 2     1     3                    6     1     3
 2     2     4                    8     0     4
 3     1     2                    6     2     0
 3     1     2                    6     2     0
 3     2     1                    6     1    -3
 4     2     2                    8     2    -2
 5     3     6                   14     2     4
 5     5     8                   18     0     6
 6     3     5                   14     3     1
 6     5     3                   14     1    -5
 7     4     7                   18     3     3
 7     7     4                   18     0    -6
 8     5     5                   18     3    -3
 8     6    10                   24     2     6
 8     7    11                   26     1     7
10     6     8                   24     4     0
10     6     8                   24     4     0
10     8     6                   24     2    -6
11     7     8                   26     4    -2
11     8     7                   26     3    -5
12    12     8                   32     0    -8
12     8    12                   32     4     4
13    10    15                   38     3     7
13    12    17                   42     1     9
15    10    13                   38     5     1
15    11    16                   42     4     6
15    13    10                   38     2    -8
15    15    20                   50     0    10