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 - http://www.scopus.com/inward/record.url?scp=74949097699&partnerID=8YFLogxK

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 -