TY - JOUR
T1 - On coincidences among quadratic fields generated by the shanks sequence
AU - Banks, William D.
AU - Shparlinski, Igor E.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - Let g>1 be an integer and f(X)εZ[X] a polynomial of positive degree with no multiple roots, and put u(n)=f(gn). In this note, we study the sequence of quadratic fields Q(√u(n)) as n varies over the consecutive integers M+1,...,M+N. Fields of this type include Shanks fields and their generalizations. Using the square sieve together with new bounds on character sums, we improve an upper bound of Luca and Shparlinski (Proc. Edinb. Math. Soc. (2) 52 (2009), 3) on the number of nε{M+1,...,M+N} with Q √u(n)=Q(√s) for a given squarefree integer s.
AB - Let g>1 be an integer and f(X)εZ[X] a polynomial of positive degree with no multiple roots, and put u(n)=f(gn). In this note, we study the sequence of quadratic fields Q(√u(n)) as n varies over the consecutive integers M+1,...,M+N. Fields of this type include Shanks fields and their generalizations. Using the square sieve together with new bounds on character sums, we improve an upper bound of Luca and Shparlinski (Proc. Edinb. Math. Soc. (2) 52 (2009), 3) on the number of nε{M+1,...,M+N} with Q √u(n)=Q(√s) for a given squarefree integer s.
UR - http://www.scopus.com/inward/record.url?scp=85021795454&partnerID=8YFLogxK
U2 - 10.1093/qmath/haw054
DO - 10.1093/qmath/haw054
M3 - Article
AN - SCOPUS:85021795454
VL - 68
SP - 465
EP - 484
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 2
ER -