The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$

Francesco Battistoni; Giuseppe Molteni; Lo\"ic Greni\'e

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The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$

Francesco Battistoni, Loïc Grenié, Giuseppe Molteni

Number Theory · Submitted 23 Jan 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$.

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arXiv ID2401.13118 (version 2)

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CitationFrancesco Battistoni, Giuseppe Molteni, Lo\"ic Greni\'e. "The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$." arXiv:2401.13118 (2024). https://pith.science/paper/T7EBLEAU

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,848 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.78 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the large deviations of $T(d)$.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $d$ be any positive and non square integer.	background 1.00	0.05	0.01	asserted 0.74
s1	We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$.	contribution 0.69	0.97	0.87	asserted 1.00
s2	This allows to improve the existing results for the large deviations of $T(d)$.	interpretation 0.94	0.75	0.15	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the large deviations of $T(d)$.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites as background 1 paper

Korolev has announced via a preprint in arXiv [ref] that $∑_d≤x g(d)^2 = (c_0 + O_(x^-1/108+)) x^2$ with $c_0 = 1.2…$.

An upper bound for the second moment of the length of the period of the continued fraction expansion for $\sqrt{d}$ · Introduction and results · 0.80

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