The octonion integers

Abstract. The real numbers, the complex numbers, the quaternions, and the octonions comprise the four composition algebras. The first three are usually seen to be the well-behaved members of the family; the octonions, to quote John Baez, are "the crazy old uncle nobody lets out of the attic." I would like to let the octonions out of the attic to introduce some of its fundamental properties, with an eye toward its most important ring of integers, the integral Cayley numbers of Coxeter. A beautiful and remarkably simple algorithm of Rehm leads to a unique factorization theorem for this ring, despite its non-associativity. Just as remarkably, there are several elementary questions about factorization in this and related rings that remain unanswered.