Hurwitz quaternion

The set of all Lipschitz quaternions forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder.

Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.

The (arithmetic, or field) norm of a Hurwitz quaternion a + bi + cj + dk, given by a2 + b2 + c2 + d2, is always an integer.

By a theorem of Lagrange every nonnegative integer can be written as a sum of at most four squares.

Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion.

More precisely, the number c(n) of Hurwitz quaternions of given positive norm n is 24 times the sum of the odd divisors of n. The generating function of the numbers c(n) is given by the level 2 weight 2 modular form where and is the weight 2 level 1 Eisenstein series (which is a quasimodular form) and σ1(n) is the sum of the divisors of n. A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units.

A Hurwitz integer is irreducible if and only if its norm is a prime number.