Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. 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 $d \geq 3$ and discriminant, when is it possible for them to represent primitively the same infinite subset of the integers?

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?

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. 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 $d \geq 3$ and discriminant, when is it possible for them to represent primitively the same infinite subset of the integers?

It has been pointed out to me by @StanleyYaoXiao that this can happen when $F_A = G$ for some $A \in \operatorname{GL}_2(\mathbb Q)\setminus \operatorname{GL}_2(\mathbb Z)$. I am particularly interested in the special case when the two binary forms are $F_T$ and $F_S$ for some $T, S \in \operatorname{M}_2(\mathbb Z)$ and $|\det S|=|\det T| > 1$.

At the moment, I don't know how to prove or disprove this statement. It might be the case that some additional conditions on $F$, $T$ or $S$ should be introduced to make it correct.

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. 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 $d \geq 3$ and discriminant, when is it possible for them to represent primitively the same infinite subset of the integers?