Just apply Pythagoras Theorem

We assume $p+q+r$ is nonnegative, otherwise just negate $p$ , $q$ and $r$ .

Firstly, by parity, if $p^2+q^2=r^2$ , then $p+q+r$ cannot be odd.

Secondly, if $p+q+r=0$ , then $p+q=-r$ . Substituting this into the first equation, we get $p^2+q^2=(p+q)^2$ , so $2pq=0$ . Therefore, either $p=0$ or $q=0$ , which is impossible since $p$ and $q$ are non-zero. Therefore, $p+q+r \geq 2$ .

We can construct 3 numbers that satisfy this: $p=3$ , $q=4$ , $r=-5$ , which is inspired by using the Pythagorean triplet $3^2+4^2=5^2$ . Therefore, the minimum value of $(p+q+r)^2=4$ .