On locally primitively universal quadratic forms

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are: 1) every primitively universal form nontrivially represents zero over every ring of p-adic integers, and 2) every almost universal form in five or more variables is almost primitively universal.