A nonnegative matrix A is primitive if for some positive integer m all entries in Am are positive. The smallest such m is called the exponent of A and written exp(A). For the class of primitive companion matrices X, we find exp(A) for certain A ∈ X. We find certain values of m for which there is an n×n primitive companion matrix (for given n) with exponent m. We also propose open problems for further research. © 2019 Rocky Mountain Mathematics Consortium.