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

Part of: Exponential sums and character sums Diophantine equations Multiplicative number theory Algebraic number theory: global fields

Published online by Cambridge University Press: 23 September 2009

Florian Luca and

Igor E. Shparlinski

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Florian Luca: Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, CP 58089, Morelia, Michoacán, Mexico; Email: (fluca@matmor.unam.mx)
Igor E. Shparlinski: Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia; Email: (igor@ics.mq.edu.au)

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Abstract

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Abstract

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