Distinct integer solutions to the equation $(x^2+y^2+z^2)/(x+y+z)=2n$

Question

I’ve been set this problem by my teacher, and had no idea where to start. I have a Diophantine equation:

$$ \frac{x^2+y^2+z^2}{x+y+z}=2n $$

$x, y, z$ and $n$ must be distinct positive integers. I’ve found solutions where there is an equality for any of $x, y$ and $z$ but I haven’t been able to for distinct positive integers, and would appreciate help.

Answer 1

There are infinitely many non-trivial solutions $x=y=z$ with $x$ even and $n=\frac{x}{2}$:

$$ \frac{x^2+y^2+z^2}{x+y+z}=\frac{3x^2}{3x}=x=2n. $$ So they cannot be unique.

Edit: The title now says distinct integer solutions (in the body it is still says unique integer solutions. In this case consider, for example, $$ (x,y,z,n)=(18,12,8,7). $$ This solves your equation in distinct integers.

Answer 2

This one is quite difficult; at least, difficult to find all primitive integer solutions and prove that we have all. Taking $a=x-n,b=y-n,c=z-n,$ we are led to $$ a^2 + b^2 + c^2 = 3n^2. $$ In order to get $\gcd(a,b,c,n)=1,$ it is necessary that all be odd.

I think I will leave in the zeros in the formulas for $a,b,c.$ They serve as spacers. $$ n = p^2 + q^2 + r^2 + s^2 $$ $$ a = p^2 + q^2 - r^2 - s^2 + 0 \, p q - 2 p r + 2 q r + 2 p s + 2 q s + 0 \, r s $$ $$ b = p^2 - q^2 + r^2 - s^2 + 2 p q - 0 \, p r + 2 q r - 2 p s + 0 \, q s + 2 r s$$ $$ c = p^2 - q^2 - r^2 + s^2 - 2 p q + 2 p r + 0 \, q r + 0 \, p s + 2 q s + 2 r s$$

In the table below, for each line just take the eight solutions $x=n \pm a, y=n \pm b,z=n \pm c.$ Some of those will have $x,y,z$ positive.

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    n       x    y    z                                         p    q    r    s 
    1 ;     2    2    2 x^2+y^2+z^2: 12 x+y+z: 6 * 2n: 12 ;     1    0    0    0
    3 ;     8    4    4 x^2+y^2+z^2: 96 x+y+z: 16 * 2n: 96 ;     0    1   -1   -1
    5 ;    12   10    6 x^2+y^2+z^2: 280 x+y+z: 28 * 2n: 280 ;     0    1    0   -2
    7 ;    18   12    8 x^2+y^2+z^2: 532 x+y+z: 38 * 2n: 532 ;     1    2    1    1
    9 ;    20   20   10 x^2+y^2+z^2: 900 x+y+z: 50 * 2n: 900 ;     0    2    2    1
    9 ;    22   16   14 x^2+y^2+z^2: 936 x+y+z: 52 * 2n: 936 ;     0    2   -1   -2
   11 ;    24   24   16 x^2+y^2+z^2: 1408 x+y+z: 64 * 2n: 1408 ;     0    1    1   -3
   11 ;    28   18   16 x^2+y^2+z^2: 1364 x+y+z: 62 * 2n: 1364 ;     3    1    0    1
   11 ;    30   12   12 x^2+y^2+z^2: 1188 x+y+z: 54 * 2n: 1188 ;     3    0   -1    1
   13 ;    30   26   20 x^2+y^2+z^2: 1976 x+y+z: 76 * 2n: 1976 ;     0    2    0   -3
   13 ;    32   24   18 x^2+y^2+z^2: 1924 x+y+z: 74 * 2n: 1924 ;     2   -2   -2   -1
   15 ;    34   32   20 x^2+y^2+z^2: 2580 x+y+z: 86 * 2n: 2580 ;     3    2    1    1
   15 ;    38   26   20 x^2+y^2+z^2: 2520 x+y+z: 84 * 2n: 2520 ;     1   -1    3   -2
   15 ;    40   22   16 x^2+y^2+z^2: 2340 x+y+z: 78 * 2n: 2340 ;     1   -3   -2   -1
   17 ;    40   30   30 x^2+y^2+z^2: 3400 x+y+z: 100 * 2n: 3400 ;     0    3   -2   -2
   17 ;    40   34   24 x^2+y^2+z^2: 3332 x+y+z: 98 * 2n: 3332 ;     4    0   -1    0
   17 ;    42   28   28 x^2+y^2+z^2: 3332 x+y+z: 98 * 2n: 3332 ;     0    3    2    2
   17 ;    46   22   18 x^2+y^2+z^2: 2924 x+y+z: 86 * 2n: 2924 ;     3   -2   -2    0
   19 ;    42   42   24 x^2+y^2+z^2: 4104 x+y+z: 108 * 2n: 4104 ;     1   -3    3    0
   19 ;    44   36   32 x^2+y^2+z^2: 4256 x+y+z: 112 * 2n: 4256 ;     0    3   -1   -3
   19 ;    48   30   30 x^2+y^2+z^2: 4104 x+y+z: 108 * 2n: 4104 ;     1    0    3   -3
   19 ;    50   30   20 x^2+y^2+z^2: 3800 x+y+z: 100 * 2n: 3800 ;     1    1    1   -4
   21 ;    46   44   34 x^2+y^2+z^2: 5208 x+y+z: 124 * 2n: 5208 ;     0    2    1   -4
   21 ;    50   40   32 x^2+y^2+z^2: 5124 x+y+z: 122 * 2n: 5124 ;     2    3    2    2
   21 ;    52   40   22 x^2+y^2+z^2: 4788 x+y+z: 114 * 2n: 4788 ;     4    2    0    1
   23 ;    52   48   34 x^2+y^2+z^2: 6164 x+y+z: 134 * 2n: 6164 ;     1   -3   -3   -2
   23 ;    54   48   24 x^2+y^2+z^2: 5796 x+y+z: 126 * 2n: 5796 ;     3   -2   -3   -1
   23 ;    58   42   24 x^2+y^2+z^2: 5704 x+y+z: 124 * 2n: 5704 ;     1   -3    3    2
   23 ;    60   36   30 x^2+y^2+z^2: 5796 x+y+z: 126 * 2n: 5796 ;     3    3    1    2
   25 ;    56   50   42 x^2+y^2+z^2: 7400 x+y+z: 148 * 2n: 7400 ;     0    3    0   -4
   25 ;    60   44   42 x^2+y^2+z^2: 7300 x+y+z: 146 * 2n: 7300 ;     4   -2   -2   -1
   25 ;    60   48   36 x^2+y^2+z^2: 7200 x+y+z: 144 * 2n: 7200 ;     1   -2    4   -2
   25 ;    66   38   30 x^2+y^2+z^2: 6700 x+y+z: 134 * 2n: 6700 ;     1    4    2    2
   25 ;    68   30   26 x^2+y^2+z^2: 6200 x+y+z: 124 * 2n: 6200 ;     2   -1    4   -2
    n      x    y    z                                                p    q    r    s 

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Answer 3

Above equation shown below: \begin{equation} (x^2+y^2+z^2)/(x+y+z)=2n \tag{A} \end{equation}

Equation $(A)$ has parametric solution:

$x=b(a+b)$

$y=a(7a+b)$

$z=2ab$

$n=(1/2)*(7a^2-2ab+b^2)$

For $(a,b)=(1,17)$ we get:

$(x,y,z,n)= (306,24,34,131)$