Primitive divisors of Lucas and Lehmer sequences

Abstract: Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor.