  In this article we compute the $q$th power values of the quadratic
polynomials $f$ with negative squarefree discriminant such that $q$ is coprime
to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory
of unique factorisation and that of primitive divisors of integer sequences is
used to deduce a bound on the values of $q$ which is small enough to allow the
remaining cases to be easily checked. The results are used to determine all
perfect power terms of certain polynomially generated integer sequences,
including the Sylvester sequence.
