Uniform estimates for primitive divisors in elliptic divisibility sequences

Abstract

Let P be a nontorsion rational point on an elliptic curve E, given by a minimal Weierstrass equation, and write the first coordinate of nP as A n a D n2, a fraction in lowest terms. The sequence of values D n is the elliptic divisibility sequence (EDS) associated to P. A prime p is a primitive divisor of D n if p divides D n, and p does not divide any earlier term in the sequence. The Zsigmondy set for P is the set of n such that D n has no primitive divisors. It is known that Z is finite. In the first part of the paper we prove various uniform bounds for the size of the Zsigmondy set, including (1) if the j-invariant of E is integral, then the size of the Zsigmondy set is bounded independently of E and P, and (2) if the abc Conjecture is true, then the size of the Zsigmondy set is bounded independently of E and P for all curves and points. In the second part of the paper, we derive upper bounds for the maximum element in the Zsigmondy set for points on twists of a fixed elliptic curve.