Size of regulators and consecutive square–free numbers

Fouvry, Étienne; Jouve, Florent

doi:10.1007/s00209-012-1035-7

Cited by 6 publications

(12 citation statements)

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“…This improves upon the result by Fouvry and Jouve [5]. They have shown that one can take θ < 7/4 in Corollary 6.…”

Section: Size Of the Fundamental Solution Of Pell Equationssupporting

confidence: 81%

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Pairs of k-free Numbers, consecutive square-full Numbers

Reuss

2012

Preprint

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We consider the error term of the asymptotic formula for the number of pairs of k-free integers up to x. Our error term improves results by Heath-Brown, Brandes and Dietmann/Marmon. We then extend our results to r-tuples of k-free numbers and improve previous results by Tsang. Furthermore, we establish an error term for consecutive square-full integers. Finally, we will show that for all θ < 3 and for almost all D, the fundamental solution ǫ D associated to the Pell equation x 2 −Dy 2 = 1 satisfies ǫ D > D θ . This improves/recovers previous results by Fouvry and Jouve. The main tool of our work is the approximate determinant method. AcknowledgmentI am very grateful to my supervisor Roger Heath-Brown who introduced me to the determinant method and its applications. I thank him for many constructive and helpful comments on my work. I would like to thank Étienne Fouvry and Florent Jouve for drawing my attention to their paper [5] which resulted in Theorem 5 below. I am also very grateful to the EPSRC 1 and to St. Anne's College, Oxford who are generously funding and supporting my doctorate's degree and this project.

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“…This improves upon the result by Fouvry and Jouve [5]. They have shown that one can take θ < 7/4 in Corollary 6.…”

Section: Size Of the Fundamental Solution Of Pell Equationssupporting

confidence: 81%

“…I thank him for many constructive and helpful comments on my work. I would like to thank Étienne Fouvry and Florent Jouve for drawing my attention to their paper [5] which resulted in Theorem 5 below. I am also very grateful to the EPSRC 1 and to St. Anne's College, Oxford who are generously funding and supporting my doctorate's degree and this project.…”

mentioning

confidence: 99%

Pairs of k-free Numbers, consecutive square-full Numbers

Reuss

2012

Preprint

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“…Since ε D (1) is a root of the Pell equation with even parameter D (1) , we deduce that T (1) is odd and the same applies to U (2) . We continue the induction process by constructing D (3) , D (4) , . .…”

Section: 2mentioning

confidence: 99%

“…In this work, we intend to pursue the study initiated by Hooley [5], and followed by the authors [1][2][3], of the statistical properties of the size of ε D . To do so, we consider for x ≥ 2 and 0 ≤ α 1 < α 2 the following set: and let S f (x; α 1 , α 2 ) be its cardinality.…”

Section: Introductionmentioning

confidence: 99%

Fundamental solutions to Pell equation with prescribed size

Fouvry

Jouve

2012

Proc. Steklov Inst. Math.

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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.

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“…The proof of this theorem is essentially based on [Fouvry and Jouve 2013] and Proposition 7. It will be given in Section 6 where we will explain why the inequality (6) is better than the trivial upper bound by some constant factor strictly larger than 3.5.…”

Section: Introductionmentioning

confidence: 99%