Does there exist a $23$ $(p = 23)$ variate (variable) polynomial $P(x, x_1, x_2..., x_{22})$ where every integer $n$ has a solution in $P(x, x_1, x_2..., x_{22})$ if and only if all divisors of $n$ are congruent to $0$ or $1$ $\pmod 23$.
For $p = 2, 3, 5, 7, 11, 13, 17, 19$ this is proven to be the case because the cyclotomic fields are PID.
Thanks for help.