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 -