Simple Functions in Generating Primes

Let \(p_n\) be the \(n\)-th prime

Are there any polynomial functions \( F(x_1, x_2, \ldots, x_k) \) in \(k\) variables such that

\[F(p_{n-1}, p_{n-2}, p_{n-3}, ... , p_{n-k}) = p_n \, \forall n >k?\]

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