Is there a notion of an octonion prime? A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime. A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is prime. I know there is an eight-square identity that underlies the octonions. Is there a parallel statement, something like: an octonion is prime if its norm is prime?
I ask out of curiosity and ignorance.
Addendum. The Conway-Smith book Bruce recommended is a great source on my question. As there are several candidates for what constitutes an integral octonion, the situation is complicated. But a short answer is that unique factorization fails to hold, and so there is no clean notion of an octonion prime. C.-S. select out and concentrate on what they dub the octavian integers, which, as Bruce mentions, geometrically form the $E^8$ lattice. Here is one pleasing result (p.113): If $\alpha \beta = \alpha' \beta'$, where $\alpha$ and $\beta$ \alpha, \alpha', \beta, \beta'$ are nonzero octavian integers, then the angle between $\alpha$ and $\alpha'$ is the same as the angle between $\beta$ and $\beta'$.
A non-serious postscript: Isn't it curious that $\mathbb{N}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ correspond to N, C, H, O, the four atomic elements that comprise all proteins and much of organic life? Water-space: $\mathbb{H}^2 \times \mathbb{O}$, methane-space: $\mathbb{C} \times \mathbb{H}^4$, ...
