Let $\alpha$ and $\beta$ are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\alpha$ and $\beta$ with h$(\beta /\alpha )\le 4$, the $n$-th element of these sequences has a primitive divisor for $n\>30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.