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**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)].

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)].

Reviewer: Andrea Bandini (Pisa)

### MSC:

11G05 | Elliptic curves over global fields |

14G05 | Rational points |

14G25 | Global ground fields in algebraic geometry |

### Citations:

Zbl 1093.11038### Software:

PARI/GP
Full Text:
DOI

### References:

[1] | Bilu, Yu.; Hanrot, G.; Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539, 75-122 (2001), (with an appendix by M. Mignotte) · Zbl 0995.11010 |

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[3] | G. Cornelissen, K. Zahidi, Complexity of undecidable formulæ in the rationals and inertial Zsigmondy theorems for elliptic curves, preprint; G. Cornelissen, K. Zahidi, Complexity of undecidable formulæ in the rationals and inertial Zsigmondy theorems for elliptic curves, preprint · Zbl 1178.11076 |

[4] | Elkies, N. |

[5] | Everest, G.; Mclaren, G.; Ward, T., Primitive divisors of elliptic divisibility sequences, J. Number Theory, 118, 71-89 (2006) · Zbl 1093.11038 |

[6] | P. Ingram, Contributions to the arithmetic of elliptic curves, PhD thesis, University of British Columbia, 2006; P. Ingram, Contributions to the arithmetic of elliptic curves, PhD thesis, University of British Columbia, 2006 |

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[8] | Silverman, J. H., The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106 (1986), Springer: Springer New York · Zbl 0585.14026 |

[9] | Silverman, J. H., Wieferich’s criterion and the abc-conjecture, J. Number Theory, 30, 226-237 (1988) · Zbl 0654.10019 |

[10] | The PARI Group, Bordeaux, PARI/GP, version 2.1.7, 2005. Available from: http://pari.math.u-bordeaux.fr/; The PARI Group, Bordeaux, PARI/GP, version 2.1.7, 2005. Available from: http://pari.math.u-bordeaux.fr/ |

[11] | Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math., 3, 265-284 (1892) |

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