Completely p-primitive binary quadratic forms

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 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. (C) 2018 Elsevier Inc. All rights reserved.