Counting fundamental solutions to the Pell equation with prescribed size

Xi, Pengcheng

doi:10.1112/s0010437x18007480

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…The first bound is an immediate consequence of [MW, (3)] (see also [La2, Theorem 1.2(a)], as well as results towards a conjecture of Hooley [Ho] due to Fouvry [Fo, Theorem 1.1] with subsequent improvements in [Bo, Xi], that further addresses the question of the density of values d with attached fundamental unit of prescribed size) in the case , and of Chowla [C] (see also [Du] for generalisations to number fields of higher degree) in the case .…”

Section: Supersolvable Extensionsmentioning

confidence: 98%

Distribution of Frobenius Elements in Families of Galois Extensions

Fiorilli¹,

Jouve

2023

J. Inst. Math. Jussieu

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Given a Galois extension $L/K$ of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem, which under GRH is sharper than the Murty–Murty–Saradha and Bellaïche refinements of the Lagarias–Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery’s conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families, these fluctuations are dominated by a constant lower order term. As an application of our ideas, we refine and generalize results of K. Murty and of Bellaïche, and we answer a question of Ng. In particular, in the case where $L/{\mathbb {Q}}$ is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein–Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques.

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Section: Supersolvable Extensionsmentioning

confidence: 98%

Distribution of Frobenius Elements in Families of Galois Extensions

Fiorilli¹,

Jouve

2023

J. Inst. Math. Jussieu

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“…Currently, there are still several important issues regarding the Pell equation. For instance, the study of the size of the fundamental solution is an interesting problem addressed in several papers, e.g., [7,11,22]. Recently, the solvability of simultaneous Pell equations and explicit formulas for their solutions have been also studied in [14,8,12].…”

Section: Introductionmentioning

confidence: 99%

On the cubic Pell equation over finite fields

Dutto¹,

Murru²

2022

Preprint

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The classical Pell equation can be extended to the cubic case considering the elements of norm one in Z[ 3 √ r], which satisfyThe solution of the cubic Pell equation is harder than the classical case, indeed a method for solving it as Diophantine equation is still missing [2].In this paper, we study the cubic Pell equation over finite fields, extending the results that hold for the classical one. In particular, we provide a novel method for counting the number of solutions in all possible cases depending on the value of r. Moreover, we are also able to provide a method for generating all the solutions.

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On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

Petit

2020

International Mathematics Research Notices

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We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

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Counting fundamental solutions to the Pell equation with prescribed size

Cited by 5 publications

References 11 publications

Distribution of Frobenius Elements in Families of Galois Extensions

Distribution of Frobenius Elements in Families of Galois Extensions

On the cubic Pell equation over finite fields

On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

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