Matteo Verzobio

Primitive divisors in elliptic divisibility sequences

Let E be an elliptic curve defined over Q and P ∈ E(Q) be a non-torsion point. Write x(nP) = An/Bn where An and Bn are two coprime integers. We want to study the sequence of integers {Bn}n∈N. We show some arithmetical properties of this sequence. In particular, we focus on the problem of understanding when a term Bn has a primitive divisor, i.e. when there exists a prime p that divides Bn but does not divide Bk for 0 < k < n. Firstly, we talk about the main results of the subject and the questions that are still open. Secondly, we present the techniques that are necessary in order to study these sequences, as the theory of heights or the diophantine approximation. Finally, we introduce a related problem: let P and Q be two points on the curve and put x(nP + Q) = Cn/Dn. We study the problem of the primitive divisors for the sequences Dn.