On quadratic fields generated by the Shanks sequence

Florian Luca; Igor E. Shparlinski

doi:10.1017/S001309150700123X

On quadratic fields generated by the Shanks sequence

Florian Luca^*, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

32 Downloads (Pure)

Abstract

Let u(n)=f(gⁿ), where g > 1 is integer and f(X)ε ℤ [X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among ℚ(√u(n)) for n ε {M+1,⋯,M+N\}. Fields of this type include the Shanks fields and their generalizations.

Original language	English
Pages (from-to)	719-729
Number of pages	11
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	52
Issue number	3
DOIs	https://doi.org/10.1017/S001309150700123X
Publication status	Published - Oct 2009

Bibliographical note

Copyright 2009 Cambridge University Press. Article originally published in Proceedings of the Edinburgh Mathematical Society, Vol. 52 No. 3, pp 719-729. The original article can be found at http://dx.doi.org/10.1017/S001309150700123X

Access to Document

10.1017/S001309150700123X

Publisher version (open access).pdfFinal published version, 175 KB

Cite this

@article{be268673adb74fb29a5e4b4098fabbc6,

title = "On quadratic fields generated by the Shanks sequence",

abstract = "Let u(n)=f(gn), where g > 1 is integer and f(X)ε ℤ [X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among ℚ(√u(n)) for n ε \{M+1,⋯,M+N\textbackslash{}\}. Fields of this type include the Shanks fields and their generalizations.",

author = "Florian Luca and Shparlinski, \{Igor E.\}",

note = "Copyright 2009 Cambridge University Press. Article originally published in Proceedings of the Edinburgh Mathematical Society, Vol. 52 No. 3, pp 719-729. The original article can be found at http://dx.doi.org/10.1017/S001309150700123X",

year = "2009",

month = oct,

doi = "10.1017/S001309150700123X",

language = "English",

volume = "52",

pages = "719--729",

journal = "Proceedings of the Edinburgh Mathematical Society",

issn = "0013-0915",

publisher = "Cambridge University Press (CUP)",

number = "3",

}

TY - JOUR

T1 - On quadratic fields generated by the Shanks sequence

AU - Luca, Florian

AU - Shparlinski, Igor E.

N1 - Copyright 2009 Cambridge University Press. Article originally published in Proceedings of the Edinburgh Mathematical Society, Vol. 52 No. 3, pp 719-729. The original article can be found at http://dx.doi.org/10.1017/S001309150700123X

PY - 2009/10

Y1 - 2009/10

N2 - Let u(n)=f(gn), where g > 1 is integer and f(X)ε ℤ [X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among ℚ(√u(n)) for n ε {M+1,⋯,M+N\}. Fields of this type include the Shanks fields and their generalizations.

AB - Let u(n)=f(gn), where g > 1 is integer and f(X)ε ℤ [X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among ℚ(√u(n)) for n ε {M+1,⋯,M+N\}. Fields of this type include the Shanks fields and their generalizations.

UR - https://www.scopus.com/pages/publications/74949097699

U2 - 10.1017/S001309150700123X

DO - 10.1017/S001309150700123X

M3 - Article

AN - SCOPUS:74949097699

SN - 0013-0915

VL - 52

SP - 719

EP - 729

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

IS - 3

ER -

On quadratic fields generated by the Shanks sequence

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this