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PRIME DIVISORS OF SEQUENCES ASSOCIATED TO ELLIPTIC CURVES

Published online by Cambridge University Press:  31 January 2005

GRAHAM EVEREST  and
IGOR E. SHPARLINSKI
GRAHAM EVEREST
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK e-mail: [email protected]
IGOR E. SHPARLINSKI
Affiliation:
Department of Computing, Macquarie University, NSW 2109, Australia e-mail: [email protected]
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Abstract

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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.

Keywords

11A4111G05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 1 , January 2005 , pp. 115 - 122
Copyright
2005 Glasgow Mathematical Journal Trust