From research completely unrelated to Number Theory I stumbled onto the following equation:


$$
xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3
$$


for $x, y, z$ integers, $x,y,z \neq 0$. Are there nonvanishing integers that satisfy it (there are many solutions if one of them is zero)?



$xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3$ in nonvanishing integers
No. This can be verified via the following Magma code, which can be used on the [free Magma online calculator](http://magma.maths.usyd.edu.au/calc):



```
P<x,y,z>:=ProjectiveSpace(Rationals(),2);
C:=Curve(P,x*y*z-7/16*((2*x-y-z)/3)^3);
E:=EllipticCurve(C);
MordellWeilGroup(E);

```

which outputs



```
Abelian Group isomorphic to Z/3
Defined on 1 generator
Relations:
    3*$.1 = 0

```

This shows that, when the given equation is viewed as a curve in $\mathbf{P}^2$, it has exactly three rational points.
One can use Magma to automatically compute these three points, or just observe that there are obviously three rational points with $xyz=0$.


