Perhap this remark at inhttps://math.stackexchange.com/questions/412980/kx-1-x-2-d‌​ots-is-a-ufd might be useful: Paul Cohn introduced the idea of inert extensions when studying Bezout rings. Cohn proved that every gcd domain can be inertly embedded in a Bezout domain, and every UFD can be inertly embedded in a PID.

Just to note that such numbers are 5-smooth or regular numbers. See smooth numbers and regular numbers in Wikipedia A formula for the number of such numbers less than n is given in the second link.

The unit quaternions form a (Lie) group isomorphic to $SU(2)$ and $Spin(3)$ which is ubiquitous in physics.

It has been a long time since we thought about this problem, so I cannot add anything to what is in the papers.