If $\{p_i\}$ is the sequence of all primes, is it possible that there exist $P\in \mathbb{Z}[x_1,\dots x_n]$ such that $P(p_i,p_{i+1},\dots p_{i+n-1})$ is bounded in $i$?

More precisely, can widely believed conjectures, or even heuristic arguments, help make such a claim unlikely.

If $\{p_i\}$ is the sequence of all primes, is it possible that there exist a non constant $P\in \mathbb{Z}[x_1,\dots x_n]$ such that $P(p_i,p_{i+1},\dots p_{i+n-1})$ is bounded in $i$?

More precisely, can widely believed conjectures, or even heuristic arguments, help make such a claim (even more) unlikely.