Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 251-274
  • ISSN: 1246-7405

Abstract

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Let α and β are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all α and β with h ( β / α ) ≤ 4 , the n -th element of these sequences has a primitive divisor for n > 30 . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

How to cite

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Voutier, Paul M.. "Primitive divisors of Lucas and Lehmer sequences, II." Journal de théorie des nombres de Bordeaux 8.2 (1996): 251-274. <http://eudml.org/doc/247835>.

@article{Voutier1996,
abstract = {Let $\alpha $ and $\beta $ are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\alpha $ and $\beta $ with h$( \beta / \alpha ) \le 4$, the $n$-th element of these sequences has a primitive divisor for $n &gt; 30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.},
author = {Voutier, Paul M.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Lucas sequence; Lehmer sequence; primitive divisor; lower bound; linear forms in logarithms},
language = {eng},
number = {2},
pages = {251-274},
publisher = {Université Bordeaux I},
title = {Primitive divisors of Lucas and Lehmer sequences, II},
url = {http://eudml.org/doc/247835},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Voutier, Paul M.
TI - Primitive divisors of Lucas and Lehmer sequences, II
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 251
EP - 274
AB - Let $\alpha $ and $\beta $ are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\alpha $ and $\beta $ with h$( \beta / \alpha ) \le 4$, the $n$-th element of these sequences has a primitive divisor for $n &gt; 30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.
LA - eng
KW - Lucas sequence; Lehmer sequence; primitive divisor; lower bound; linear forms in logarithms
UR - http://eudml.org/doc/247835
ER -

References

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