## Elliptic divisibility sequences over certain curves.(English)Zbl 1170.11010

A divisibility sequence is a sequence of integers $$C=\{C_n\}_{n\geq 1}$$ such that $$C_n\mid C_m$$ whenever $$n\mid m$$. The paper deals with divisibility sequences arising from non-torsion rational points of an elliptic curve $$E$$: namely if $$P\in E(\mathbb{Q})$$ is a point of infinite order, one puts $$x(nP)=\frac{A_n}{B_n}$$ (in reduced form and with $$B_n>0$$) and $$B(E,P)=\{B_n\}_{n\geq 1}$$ is an elliptic divisibility sequence. A prime divisor $$p$$ of $$B_n$$ is called a primitive divisor if $$p\nmid B_m$$ for any $$m<n$$. The goal of the paper is to give informations (and upper bounds) for the Zsigmondy bound $$Z(B(E,P))=\sup\{m: B_m$$ has no primitive divisor$$\}$$ when $$E$$ has $$j$$-invariant 0 or 1728.
Mainly by direct computations the author shows that if $$B_n$$ has no primitive divisor then $$x(P)=\frac{a}{b^2}$$ is associated to a solution of an explicit system of Thue equations depending on the division polynomials of the elliptic curve. Solving these equations for a fixed $$n$$ provides the list of all curves $$E$$ and points $$P$$ such that $$B_n$$ has no primitive divisor.
In the case of $$E: y^2=x^3-N^2x$$ with squarefree $$N$$ these computations allow the author to sharpen some of the bounds obtained by G. Everest, G. Mclaren and T. Ward [J. Number Theory 118, No. 1, 71–89 (2006; Zbl 1093.11038)].

### MSC:

 11G05 Elliptic curves over global fields 14G05 Rational points 14G25 Global ground fields in algebraic geometry

Zbl 1093.11038

PARI/GP
Full Text:

### References:

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