Fundamental solutions to Pell equation with prescribed size

Abstract

We prove that the number of parameters D up to a fixed x ≥ 2 such that the fundamental solution ε D to the Pell equation T 2 − DU 2 = 1 lies between D 1 2 +α1 and D 1 2 +α2 is greater than √ 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 D ≤ x for which log ε D is larger than D 1 4 has a cardinality essentially larger than x 1 4 log 2 x.

Counting fundamental solutions to the Pell equation with prescribed size

“…for all large x > x 0 (α, β). This weaker statement was made unconditionally by Fouvry and Jouve [FJ12] whenever β < 3 2 . It is expected that the arguments in this paper can enlarge the admissible range of β.…”

“…A weaker statement would assert that, for any there exists a positive constant , such that for all large . This weaker statement was made unconditionally by Fouvry and Jouve [FJ12] whenever . It is expected that the arguments in this paper can enlarge the admissible range of .…”

Approximation diophantienne et distribution locale sur une surface torique II

“…Il a des similarités avec la méthode de «descente» qui remonte à Legendre et Dirichlet (cf. par exemple [24] pour un survol historique et les références dedans) et est récemment reprises par Fouvry et Jouve dans les travaux [10], [11] et [12] pour traiter la solution fondamentale des équations de Pell-Fermat x 2 − Dy 2 = 1.…”