Are there any non-trivial rational integers in the $p$-adic closure of $\{1,q,q^2,q^3,...\}$? If $p$ is prime and not a divisor of $q$, are there any non-trivial rational integers in the $p$-adic closure of the set or powers of $q$? Edit: $q$ is also a (rational) integer, not a $p$-adic.

If $p>2$ then there is an integer $q$ not divisible by $p$, with the properties: $q^k\
ot\equiv 1$ mod $p$ for $1\leq k\leq p-2$ and $q^{p-1}\
ot\equiv 1$ mod $p^2$. Under these conditions the $p$-adic closure of $\\{1,q,q^2,\dots\\}$ is the whole $\mathbb{Z}_p^\times$ - in particular it contains all the rational integers not divisible by $p$.

For $p=2$ and $q\equiv 5$ mod $8$ then the closure is $1+4\mathbb{Z}_2$ - i.e. it contains all the integers which are $1$ mod $4$.