  Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined
over a number field $K$. In this paper, we study the denominators of the
$x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of
prime order, then for $n$ large enough there always exists a primitive divisor.
Later on, we show the link between the study of the primitive divisors and the
Lang-Trotter conjecture.
