Let $x,y,z$ integers and $p$ is a prime satisfying
$x^3 + y^3 + z^3 \equiv 0 \pmod p \Rightarrow xyz \equiv 0 \pmod p$
this proposition. Is there infinity $p$ primes or $p$ is only $2,7,13$ ?
Answer is easy when
$x^2 + y^2 + z^2 \equiv 0 \pmod p \Rightarrow xyz \equiv 0 \pmod p$
Only $2$ and $5$ satisfies this proposition. We can use Legendre symbol or other elementary methods.
For $x^3$, there is a tool like a legendre symbol. But i don't know how to use it.
For $p\ge5$, $X^3+Y^3+Z^3=0$ is the homogeneous equation of an elliptic curve in the projective plane. It has $p+1-a$ points over $\Bbb F_p$ where $|a|<2\sqrt p$ (Hasse's theorem). But it has most $9$ points with $XYZ=0$. As long as $p+1-2\sqrt p=(\sqrt p-1)^2>9$, it has points with $XYZ\ne0$. This is the case for $p\ge17$.