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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.

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    Less trivially, (p-q-2)(p-q-4)(p-q-6)(p-q-8)(p-q-10)(p-q-12)(p-q-14)(p-q-16)(p-q-18)(p-q-20) is zero infinitely often under Elliott-Halberstam.2012-02-09

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Meanwhile, we know the answer is yes. As of 2013 we know (thanks to Y. Zhang) that there is some $C\leq7\cdot10^7$ such that there are infinitely many prime pairs that differ by $C$. So we can take $R(x,y)=x-y-C$, or, if you want an explicit example, take $\prod_{\substack{C=2\\C\text{ even}}}^{7\cdot10^7}(x-y-C)$. As of april 2014 it is known that we can have $C\leq246$.

See here for more information.