Let $F(x,y)$ be an irreducible binary form with integer coefficients, degree $d \geq 3$ and content 1. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ and $y$ such that $F(x,y)=n$. We also say that a subset $\mathcal S \subset \mathbb Z$ is primitively represented by $F$ if every $n \in \mathcal S$ is primitively represented by $F$.

Next, for any matrix

$$A = \left(\begin{matrix}a & b\\c & d\end{matrix}\right)$$

with rational entries, define

$$F_A(x,y) = F(ax + by, cx + dy).$$

We say that two binary forms $F$ and $G$ are *equivalent* if there exists $A \in \operatorname{GL}_2(\mathbb Z)$ such that $F_A = G$.

I am interested in the following question: *given two distinct non-equivalent irreducible binary forms $F$ and $G$ of the same degree $d \geq 3$, same discriminant and content 1, when is it possible for them to represent primitively the same infinite subset of the integers?*

nothold? $\endgroup$ – Daniel Loughran Jun 30 '16 at 12:14