WillJagy answered a linear relation question on Pythagorean Triples in [Small linear relations between primitive Pythagorean triples $\mathsf I$](https://mathoverflow.net/questions/357219/small-linear-relations-between-primitive-pythagorean-triples).


Now let $a^2+b^2=c^2$ be a primitive Pythagorean triple and then consider the Linear Diophantine Equation $ua+vb+zc=0$ where $(u,v,z)\in\mathbb Z^3$ are variables. If $(u,v,z)\neq(0,0,0)$ then:


1. Is $\|(u,v,z)\|\_\infty$ at least $\sqrt{\max(|a|,|b|)}$ up to constant factors or should the scale (disregarding constants) be smaller (perhaps $\sqrt[3]{\max(|a|,|b|)}$)?
2. What is the distribution of $\|(u,v,z)\|\_\infty$?


Note if it were $ua^2+vb^2+zc^2=0$ then the answer is $O(1)$ since $(u,v,z)=(1,1,-1)$ suffices.




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This is what I have
$$ a = m^2 - n^2 $$
$$ b = 2mn $$
$$ c = m^2 + n^2 $$
then
$$ n(m^2 - n^2 ) +(-m)(2mn) + n(m^2 + n^2) = 0 $$
or triple
$$(u,v,z)=(n,-m,n)$$ works and this gives morally $\sqrt{\max(|a|,|b|)}$ ($(m,n,-m)$ also works to give morally $\sqrt{\max(|a|,|b|)}$). Could there be something smaller?



Small linear relations between primitive Pythagorean triples $\mathsf{II}$
1. Yes, the minimal $\|(u,v,z)\|\_\infty$ is within a constant factor of
$\sqrt{|c|}$ (equivalently, of $\sqrt{\max(|a|,|b|)}$.


The orthogonal complement of $(a,b,c) = (m^2-n^2, 2mn, m^2+n^2)$
contains the independent integer vectors 
$v\_1 := (n,-m,n)$ (which you found) and $v\_2 := (m,n,-m)$.
Their $\bf Z$-span is the full integral complement of $(a,b,c)$,
for example because that span has discriminant
$$
\|v\_1\|\_2^2 \|v\_2\|\_2^2 - \langle v\_1, v\_2 \rangle^2
= 2(m^2+n^2)^2 
$$
which equals $a^2+b^2+c^2 = \|(a,b,c)\|\_2^2$.
Moreover, $(v\_1,v\_2)$ is a reduced basis, because
$|\langle v\_1, v\_2 \rangle| = mn$ is less than $(m^2+n^2)/2$, 
and thus less than both $ \frac12 \|v\_1\|\_2^2$ and $\frac12 \|v\_2\|\_2^2$.
Therefore the minimal $\|(u,v,z)\|\_2$ exceeds 
$\sqrt{m^2+n^2} = \sqrt{|c|}$, whence the same is true
of $\|(u,v,z)\|\_\infty$ within a constant factor. **QED**


