ABSTRACT
One can obtain an interesting ring by reading the octavian integers modulo any prime p (just as when reading the ordinary integers modulo p one obtains an interesting ring that is, in fact, a field). To be precise, one says that two octavian integers are congruent mod p provided that their difference is p times an octavian integer. When we study the lattice of octavian integers mod 2, it will be important to count those of various norms. The octavian integers form a scaled copy of the E8 root lattice, and it is known that the number of vectors of any norm n > 0 is 240 times the sum of the cubes of the divisors of n. The chapter also proves that, up to symmetry, the only subrings modulo 2 of O8 are the unit-rings, taken modulo 2, and four others.
