Uniform estimates for primitive divisors in elliptic divisibility sequences

Patrick Ingram () and Joseph H. Silverman ()
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Patrick Ingram: Colorado State University, Department of Pure Mathematics
Joseph H. Silverman: Brown University, Mathematics Department

A chapter in Number Theory, Analysis and Geometry, 2012, pp 243-271 from Springer

Abstract: 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 ∕ D n 2, 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.

Keywords: elliptic divisibility sequence; elliptic curve; primitive divisor (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-1260-1_12

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DOI: 10.1007/978-1-4614-1260-1_12

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