ABSTRACT
This chapter deals with Hurwitz integral quaternions. How shall we define what it means for the quaternion q = a + bi + cj + dk to be integral? One possibility is to demand that a, b, c, and d be ordinary rational integers. Hurwitz later found a different definition that turns out to have nicer properties. We say that a+bi+cj+dk is a Hurwitz integer just if either all of a, b, c, d are in ℤ or all are in ℤ+12. The chapter proves that Hurwitz integers, but not the Lipschitz ones, have the “division with small remainder” property. A prime Hurwitz integer P is one whose norm is a rational prime p. Analogously to the fact that p = p × 1 and p = 1 × p are the only ways p is the product of two rational primes. The chapter examines the quaternionic factorizations of rational integers and, in particular, of a rational prime p.
