Title: Completely $p$-primitive binary quadratic forms,
Abstract: Let $f(x,y)=ax^2+bxy+cy^2$ 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 $f(x,y)=N$ has always 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.