Abstract

Elliptic divisibility sequences arise as sequences of denominators of the integer multiples of a rational point on an elliptic curve. Silverman proved that almost every term of such a sequence has a primitive divisor (that is, a prime divisor that has not appeared as a divisor of earlier terms in the sequence). If the elliptic curve has complex multiplication, then we show how the endomorphism ring can be used to index a similar sequence and we prove that this sequence also has primitive divisors. The original proof fails in this context and will be replaced by an inclusion-exclusion argument and sharper diophantine estimates.

Keywords

complex multiplication, divisibility sequence, elliptic curve, endomorphism, primitive divisor, Zsigmondy

Mathematical Subject Classification 2000

Primary: 14H52

Secondary: 14K22

Milestones

Received: 30 July 2007
Revised: 12 November 2007
Accepted: 25 December 2007

Authors