Some effectivity results for primitive divisors of elliptic divisibility sequences

Abstract

Let P be a nontorsion point on an elliptic curve defined over a number field K and consider the sequence {B n } n∈‫ގ‬ of the denominators of x(nP). We prove that every term of the sequence of the B n has a primitive divisor for n greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.Silverman [1988, Proposition 10] proved that, if E is defined over ‫,ޑ‬ then B n has a primitive divisor for n large enough. This result was generalized for every number field K in [Cheon and Hahn 1999], where the following theorem is proved.Theorem 1.1 [Cheon and Hahn 1999, main theorem]. Let E be an elliptic curve defined over a number field K and let P be a nontorsion point in E(K ). Consider the sequence {B n } n∈‫ގ‬ of integral O K -ideals as defined in (1). Then B n has a primitive divisor for all but finitely many n ∈ ‫.ގ‬ MSC2020: 11B39, 11G05, 11G50.