Polynomial that indicates whether or not $x = 1 \pmod n$.

For each $n$, is there a polynomial that takes representatives $x \in \Bbb{Z}$ for $\bar{x} \in \Bbb{Z}_n$, and returns whether or not $x = 1 \pmod n$?

For example, $n = 2$. Then $x \mapsto x \pmod{2}$ works. Is there a general formula for the $n$th modulus?

I'm stuck on case $n= 3$.

Let's loosen the criteria, we can use any of the classical elementary functions. So exponentiation will work:

$$ n = 3: \\ f(x) = 2^x - 1 \pmod 3\\ 0 \to 2^0 - 1 = 1 - 1 = 0 \pmod 3\\ 1 \to 2^1 - 1 = 1 \pmod 3\\ 2 \to 2^2 - 1 = 3 = 0 \pmod 3 $$

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