Is every vector in $\mathbb Z^3$ a cross product?

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.

• A basic idea would be to take the cross product of two arbitrary vectors and see what you can conclude from the formulas for the components of the result Commented Apr 16, 2022 at 1:31
• I see the category "Number Theory" was removed from this question. But I used the answer to solve diophantine equations so I'm adding that category back, Commented Apr 20, 2022 at 20:02
• Predrag3141, for what it's worth, I also think that number theoretic aspects are relevant here. It may be that the tag elementary-number-theory is a better match, as the number theory is not very deep. Let's see what others think. Commented Apr 20, 2022 at 20:10
• There is a pretty deep follow-up where this is used to solve integer relations, which are the Diophantine equations I alluded to. The question itself is also a rather difficult Diophantine equation to solve though the solution is not difficult to understand. Commented Apr 20, 2022 at 20:16

Let us write $$v=(a, b, c)$$, and consider the three vectors $$w_1=(0, c, -b), \quad w_2=(-c,0,a), \qquad w_3=(b, -a, 0).$$ Note that $$w_1 \times w_2=cv$$, $$w_1 \times w_3=-bv$$. This way, let $$d=\gcd(b, c)$$, and write it as $$d=\lambda b+\mu c$$, so that $$\frac{w_1}{d} \times\left( \mu w_2-\lambda w_3 \right)=v.$$ Here $$\mu w_2-\lambda w_3$$ and $$w_1/d$$ have integer coordinates, as we can easily check.
• I think the proof works if $b=0$ and $c\neq 0$, as then we can take $d=c$, $\lambda=0$ and $\mu=1$. And by symmetry, it works provided that one of the $a, b, c$ is nonzero, and so the only special case is $(0,0,0)$. (Of course, we can check separatedly for $(0, b, c)$ and similar, as you pointed out.) Commented Apr 16, 2022 at 3:33