Is there a non zero polynomial $R \in \mathbb{Z}[X,Y]$ such that there exists an infinite number of pair $(p,q)$ with $p$ and $q$ primes, $p \neq q$ and $R(p,q)=0$ ?
I know the curve must be of genus $0$ (Faltings-Mordell).
My question is related to Polynomial equations in $n$ and $\phi(n)$ that has been solved.