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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?

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    $\begingroup$ Could you please clarify your question? Let $P_F$ (resp. $P_G$) be the set of primes primitively represented by $F$ (resp. $G$). Are you asking whether it is possible for $P_F = P_G$, or whether it is possible for $P_F \cap P_G$ to be infinite? $\endgroup$
    – Daniel Loughran
    Commented Jun 30, 2016 at 8:48
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    $\begingroup$ @DanielLoughran the condition that $P_F$ and $P_G$ consist of primes is too restrictive. Instead, let those two sets contain all numbers primitively represented by F and G, respectively. I'm asking under which conditions $P_F \cap P_G$ can be infinite. $\endgroup$
    – Anton
    Commented Jun 30, 2016 at 11:28
  • $\begingroup$ Thanks for clarifying. Do you know examples of $F$ and $G$ where this property does not hold? $\endgroup$
    – Daniel Loughran
    Commented Jun 30, 2016 at 12:14
  • $\begingroup$ @DanielLoughran I do know an example when this is not true. Stanley Yao Xiao gave an example of two binary forms of degree 4 with coefficient vectors F=(2401,98,3,4,5) and G=(1,2,3,196,12005). They are not equivalent and don't have any GL2(Q)-automorphisms. However, they are connected by means of the relation F(x,7y)=G(7x,y), and because of that there are infinitely many integers which can be primitively presented by both forms. $\endgroup$
    – Anton
    Commented Jun 30, 2016 at 12:34
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    $\begingroup$ If $d>4$ you could try applying the Bombieri-Lang conjecture to the projective surface defined by $F(x,y)=G(z,w)$ and see what you get. $\endgroup$
    – Felipe Voloch
    Commented Jul 3, 2016 at 7:08

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