  A major problem in knot theory is to decide whether the Jones polynomial
detects the unknot. In this paper we study a weaker related problem, namely
whether the Jones polynomial reduced modulo an integer $n$ detects the unknot.
The answer is known to be negative for $n=2^k$ with $k\geq 1$ and $n=3$. Here
we show that if the answer is negative for some $n$, then it is negative for
$n^k$ with any $k\geq 1$. In particular, for any $k\geq 1$, we construct
nontrivial knots whose Jones polynomial is trivial modulo~$3^k$.
