Gaussian primes, quaternion primes, ... octonions? 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
$\alpha, \alpha', \beta, \beta'$ 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: $\mathbb{C} \times \mathbb{H}^4$, ...
 A: There may be Octonion primes as Conway suggests as there are
Hurwitz and Gaussian primes, the great difficulty is that of calculating 
and visualizing an eight vector of Integers so that their Norm is 
a prime on current desktop computer. Just going 5 deep involves
looking at 5^8 numbers. Projecting the two Gaussian prime like
four vectors in 3d gives each Octonion prime two vectors of Hurwitz 
integers. The good news is that with combinations of 
a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2
eight numbers with their negatives:
{a,b,c,d,e,f,g,h,-a,-b,-c,-d,-e,-f,-g,-h}
There is a disk of Binomial[2*n,16] combinations
that make finding primes more likely. I was combinatorial
augmentation for this effect.
A: You should probably read "On Quaternions and Octonions" by J.H. Conway and D.A. Smith
P.S. It's "octonion" not "octonian"
Edit: The first thing you will find is a discussion of integral numbers. For the complex numbers you have $\mathbb{Z}[i]$ (aka Gaussian integers) which is an $A_1\times A_1$ lattice. You also have $\mathbb{Z}[\omega]$ (aka Eisenstein integers) which is an $A_2$-lattice. For quaternions you have the integral quaternions (as above) which is an $A_1^4$ lattice and you also have the Hurwitz integral quaternions (adjoin $(1+i+j+k)/2$) which as a lattice is $D_4$. The Hurwitz numbers have "division with small remainder" property which makes them better. For the octonions you might start with doubling the Hurwitz integers to get a $D_4\times D_4$ lattice. Then you might add more and get $D_8$. However the octavian integers are an $E_8$ lattice. They are not unique (there are seven versions). These have several good properties:
Every left or right ideal is principal.
Every ideal is two-sided.
Then there is a discussion of prime factorisations.
Finally the automorphism group of the octaves has a simple subgroup of index 2 and order 12096. This group is $G_2(2)$.
