Power values of certain quadratic polynomials

Anthony Flatters[1]

  • [1] School of Mathematics University of East Anglia Norwich NR4 7TJ, UK

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 645-660
  • ISSN: 1246-7405

Abstract

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In this article we compute the q th power values of the quadratic polynomials f ∈ ℤ [ x ] with negative squarefree discriminant such that q is coprime to the class number of the splitting field of f over ℚ . The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of q which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.

How to cite

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Flatters, Anthony. "Power values of certain quadratic polynomials." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 645-660. <http://eudml.org/doc/116425>.

@article{Flatters2010,
abstract = {In this article we compute the $q$th power values of the quadratic polynomials $f\in \mathbb\{Z\}[x]$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb\{Q\}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.},
affiliation = {School of Mathematics University of East Anglia Norwich NR4 7TJ, UK},
author = {Flatters, Anthony},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Primitive divisor; Diophantine equation; Lucas sequence; primitive divisor; Sylvester sequence},
language = {eng},
number = {3},
pages = {645-660},
publisher = {Université Bordeaux 1},
title = {Power values of certain quadratic polynomials},
url = {http://eudml.org/doc/116425},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Flatters, Anthony
TI - Power values of certain quadratic polynomials
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 645
EP - 660
AB - In this article we compute the $q$th power values of the quadratic polynomials $f\in \mathbb{Z}[x]$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.
LA - eng
KW - Primitive divisor; Diophantine equation; Lucas sequence; primitive divisor; Sylvester sequence
UR - http://eudml.org/doc/116425
ER -

References

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