 4

The integer $n=-3$ gives $n^3-5=-32$, whose 2-adic norm is $2^{-5}<2^{-3}$. For the 5-adic condition, if $5\nmid n$ then $n^3-5$ is a unit, while if $5\mid n$ its valuation is exactly one. Neither case has norm below $5^{-3}$. The Newton construction for the remaining requests follows.