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Fundamental solutions to Pell equation with prescribed size

  • Étienne FouvryEmail author
  • Florent Jouve
Article

Abstract

We prove that the number of parameters D up to a fixed x ≥ 2 such that the fundamental solution ɛ D to the Pell equation T 2DU 2 = 1 lies between \(D^{\tfrac{1} {2} + \alpha _1 }\) and \(D^{\tfrac{1} {2} + \alpha _2 }\) is greater than \(\sqrt x \log ^2 x\) up to a constant as long as α 1 < α 2 and α 1 < 3/2. The starting point of the proof is a reduction step already used by the authors in earlier works. This approach is amenable to analytic methods. Along the same lines, and inspired by the work of Dirichlet, we show that the set of parameters Dx for which log ɛ D is larger than D ¼ has a cardinality essentially larger than x ¼ log2 x.

Keywords

STEKLOV Institute Fundamental Solution Congruence Condition Continue Fraction Expansion Principal Character 
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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématique, UMR 8628, CNRSUniversité Paris-SudOrsayFrance

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