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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