On coincidences among quadratic fields generated by the shanks sequence

William D. Banks*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)465-484
Number of pages20
JournalQuarterly Journal of Mathematics
Issue number2
Publication statusPublished - 1 Jun 2017
Externally publishedYes


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