Vol. 2, No. 2, 2008

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11, 1 issue

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Divisibility sequences for elliptic curves with complex multiplication

Marco Streng

Vol. 2 (2008), No. 2, 183–208

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.

complex multiplication, divisibility sequence, elliptic curve, endomorphism, primitive divisor, Zsigmondy
Mathematical Subject Classification 2000
Primary: 14H52
Secondary: 14K22
Received: 30 July 2007
Revised: 12 November 2007
Accepted: 25 December 2007
Published: 15 March 2008
Marco Streng
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
The Netherlands