Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(0,0,0,0,0)$ then can we say anything about $\|(u,v,x, y, z)\|\_\infty$ or the probability distribution of $\|(u,v,x, y, z)\|\_\infty$?


By this I mean can $\|(u,v,x, y, z)\|\_\infty$ be much smaller than $\sqrt{\max(a^2,b^2)}$?



Small linear relations between primitive Pythagorean triples $\mathsf I$
Yes, when $m>n>0$ and
$$ a = m^2 - n^2 $$
$$ b = 2mn $$
$$ c = m^2 + n^2 $$
then
$$ -n a^2 +(m-n)b^2 - n ab +(m-n)bc - n ca = 0 $$
or quintuple
$$ -n, m-n, -n, m-n, -n $$


There is a second pattern that gives the same optimum when $n$ is small, quintuple
$$ -2n, m-n, m-n, m-n, -2n $$


