2 problems

For $n\geq 1$, let $H(n)$ have vertex set $a+ib$ with $1\leq a,b\leq n$, joining $a+ib$ and $c+id$ when $a+ib+jc+kd$ is a Lipschitz prime, equivalently when $a^2+b^2+c^2+d^2$ is pr…

Fix an integer $a$, and consider the hyper-plane $z=a$ in the space of Hurwitz integers. Let $r$ denote the radius of a ball and count the Hurwitz primes lying in that hyper-plane…