On primitively universal quadratic forms

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.