Completely p-primitive binary quadratic forms

Let f(x,y)=ax2+bxy+cy2 be a binary quadratic form with integer coefficients. For a prime p not dividing the discriminant of f, we say f is completely p-primitive if for any non-zero integer N, the diophantine equation f(x,y)=N always has an integer solution (x,y)=(m,n) with (m,n,p)=1 whenever it has an integer solution. In this article, we study various properties of completely p-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form f to be completely p-primitive.