Glasgow Mathematical Journal

Table of Contents - 2005 - Volume 47, Issue 01  


PRIME DIVISORS OF SEQUENCES ASSOCIATED TO ELLIPTIC CURVES

GRAHAM EVEREST a1 and IGOR E. SHPARLINSKI a2
a1 School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK e-mail: g.everest@uea.ac.uk
a2 Department of Computing, Macquarie University, NSW 2109, Australia e-mail: igor@comp.mq.edu.au

Article author query
everest g   [Google Scholar] 
shparlinski ie   [Google Scholar] 
 

Abstract

We consider the primes which divide the denominator of the $x$-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.

(Received February 10 2004)
(Accepted June 22 2004)

Maths Classification

11A41; 11G05.