On the negative Pell equation

Fouvry, Étienne; Klüners, Jürgen

doi:10.4007/annals.2010.172.2035

Cited by 51 publications

(90 citation statements)

References 32 publications

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“…What follows already appears in [3] and [4] and is a slight modification of [4, §2.2]. This illustrates the theory of moments.…”

Section: From Theorem 3 To Theoremsupporting

confidence: 68%

“…The preceding work [4] was motivated by an important paper of Stevenhagen [23], where the author constructed a clever and convincing probabilistic model to guess the answer to the question (5). His investigations led him to enunciate Conjecture 1.…”

Section: For Instance)mentioning

confidence: 99%

“…Definition of characters and symbols. All the following tools are the cornerstone of [4]. They have their origin in several papers of Redei, Scholz, Reichardt, Lemmermeyer ([13], [15], [16], [18], [19], [21],... ).…”

Section: Lemma 4 ([4 Lemmata 5 and 7])mentioning

confidence: 99%

“…We take a solution from Lemma 6. We can assume y, z > 0 and by choosing the sign of x we can get: A straightforward computation (see the proof of Lemma 20 in [4], in the particular case D is special) shows that…”

Section: éTienne Fouvry and Jürgen Klünersmentioning

confidence: 99%

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The parity of the period of the continued fraction of d

Fouvry

Klüners

2010

Proceedings of the London Mathematical Society

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Abstract. We prove that, asymptotically, in the set of squarefree integers d, not divisible by primes congruent to 3 mod 4, the period of the expansion of √ d in continued fractions is more frequently odd than even. Statement of the resultsThe subject of the expansion of the real numbers in simple continued fractions remains a very opaque domain in the theory of numbers. One of the very few achievements of this theory is the following famous theorem due to Lagrange (see [22, Theorem 3 p. 317], for instance). With the classical notations, we have the equalityIn that formula, s = s(d) is the period of the expansion, and we have the equality a t = a s−t , for any 1 ≤ t ≤ s − 1 and also the inequality s < 2d. Hence Theorem A defines an application s : d → s(d), from the set of non square integers to N * . The image of s is equal to N * (see [22, Theorem 6 p.325]) and more precisely for every positive integer s 0 , the equation s(d) = s 0 has infinitely many solutions in d.Recall that if the real number α has an ultimately periodic expansion in continued fractions, then α is an algebraic number of degree 2 (see [22, p.328] for instance).The application s is very mysterious at many points of view. Here we shall be concerned by the frequency of the odd values of this function. We recall a very useful link between the parity of s(d) and the associated negative Pell equation.

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“…What follows already appears in [3] and [4] and is a slight modification of [4, §2.2]. This illustrates the theory of moments.…”

Section: From Theorem 3 To Theoremsupporting

confidence: 68%

Section: For Instance)mentioning

confidence: 99%

Section: Lemma 4 ([4 Lemmata 5 and 7])mentioning

confidence: 99%

Section: éTienne Fouvry and Jürgen Klünersmentioning

confidence: 99%

See 2 more Smart Citations

The parity of the period of the continued fraction of d

Fouvry

Klüners

2010

Proceedings of the London Mathematical Society

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“…3,4]: 18) where C r,K and C r,ℓ,K are positive constants, given explicitly in [14,Thm. 3,4] as infinite products.…”

Section: Higher Genusmentioning

confidence: 99%

A Polynomial Analogue of Landau's Theorem and Related Problems

Gorodetsky

2017

Mathematika

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Recently, an analogue over Fq[T ] of Landau's Theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf [10]. They counted the number of monic polynomials in Fq[T ] of degree n of the form A 2 + T B 2 , which we denote by B(n, q). They studied B(n, q) in two limits: fixed n and large q, and fixed q and large n. We generalize their result to the most general limit q n → ∞. More precisely, we provefor an explicit constant Kq = 1 + O 1 q . Our methods are different and are based on giving explicit bounds on the coefficients of generating functions. These methods also apply to other problems, related to polynomials with prime factors of even degree.

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Asymptotics of Class Numbers for Real Quadratic Fields

Raulf

2021

Association for Women in Mathematics Series

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On the negative Pell equation

Cited by 51 publications

References 32 publications

The parity of the period of the continued fraction of d

The parity of the period of the continued fraction of d

A Polynomial Analogue of Landau's Theorem and Related Problems

Asymptotics of Class Numbers for Real Quadratic Fields

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