# How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $$p + 1$$, but I'm not quite sure.

Depends on what you mean by "quaternions" and "distinct". If you mean quaternions $$a + b{\bf i} + c{\bf j} + d{\bf k}$$, with $$a,b,c,d$$ all integers, then you're asking to count solutions of $$a^2+b^2+c^2+d^2 = p$$, which by Jacobi's four squares theorem is $$8$$ times the sum of positive divisors of $$p$$, i.e. $$8(p+1)$$. If you then identify quaternions $${\bf q}$$ with $${\bf qu}$$ or $${\bf uq}$$ (but not both!) where $$\bf u$$ is one of the eight units $$\pm 1, \pm{\bf i}, \pm{\bf j}, \pm{\bf k}$$, then there are $$p+1$$ distinct solutions as desired.