Is every $3$-dimensional vector $v$ with integer coordinates a cross product of two other vectors with integer coordinates?
I have written a program to check for $v$ with entries between $-7$ and $7$. Every $v$ that small can be expressed as a cross product of two other vectors with integer coordinates.
But I can't come up with a general proof.
Apart from the empirical evidence from my experiment on small $v$, another reason to think this is true is that it's almost enough to find two small independent vectors, $u$ and $w$, with integer coordinates that are perpendicular to $v$. Playing around with integer relation algorithms has taught me that such $u$ and $w$ should be plentiful. The cross product of $u$ and $w$ is a scalar multiple of $v$ - call it $kv$. $k$ is an integer; in most cases $|k| = 1$. If it isn't, pick a different $u$ and $w$.
A similar but much easier question was this: Is every vector in $\Bbb R^3$ a cross product?.
Note: The answer given here was used to solve Diophantine equations so the question is about number theory.