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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: $\mathbb{C} \times \mathbb{H}^4$, ...

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Hehe, the observation in your postscript may indeed have deeper resonances ;P –  Jose Brox Jul 6 '10 at 13:09
But what if you denote the quaternions by \mathbb{K}, like Mumford in <i>Abelian Varieties</i>? Then you need to fit potassium into your argument somehow... –  Artie Prendergast-Smith Jul 6 '10 at 14:08
And, while we're at it, we can associate two of the Conway sporadic groups with carbon dioxide and the carbonate ion. –  DavidLHarden May 5 '11 at 18:38
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2 Answers

up vote 13 down vote accepted

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 $G_2(2)$.

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That misspelling (now corrected) may have obfuscated my searches--Thanks! –  Joseph O'Rourke Jul 6 '10 at 12:41
I have requested the Conway-Smith book through Interlibrary Loan. I would be grateful if anyone cares to presage what I will find there concerning octonion primes. –  Joseph O'Rourke Jul 6 '10 at 14:17
Unfortunately now whenever I will read octonion, I will always pronounce it "oct-onion" in my head. –  Peter Humphries Jul 6 '10 at 14:49
@Bruce: Thanks for adding all that fascinating detail! Very much appreciated! –  Joseph O'Rourke Jul 6 '10 at 15:15
@Peter: You may have started a dangerous meme. :-) Perhaps try to mentally substitute Bruce's "octavian," at least in adjectival usage. –  Joseph O'Rourke Jul 6 '10 at 15:17
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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.

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