# Completely $p$-primitive binary quadratic forms

## TL;DR

This paper investigates the properties of completely p-primitive binary quadratic forms, providing a necessary and sufficient condition for definite forms to possess this property, which relates to solutions of certain Diophantine equations.

## Contribution

It introduces the concept of completely p-primitive forms and characterizes when a definite binary quadratic form is completely p-primitive.

## Key findings

- Characterization of completely p-primitive forms for definite cases
- Necessary and sufficient conditions established
- Insights into solutions of Diophantine equations related to these 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.