Yann Bugeaud, Florian Luca
In this paper, we prove that the period of the continued fraction expansion of ${\sqrt {2^{n}+1}}$ tends to infinity when $n$ tends to infinity through odd positive integers.
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Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin's probability law. Results are given for sequences of partial quotients of $\sqrt[3]{m}$ and $\sqrt[3]{m^2}$ with $m$ noncube. A big partial quotient in one sequence finds a connection in the other.
We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = η\cdot\mathfrak o(\sqrt{-3})$, $η= \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ …
For a given quadratic irrational $α$, let us denote by $D(α)$ the length of the periodic part of the continued fraction expansion of $α$. We prove that for a positive integer $d$, which is not a perfect square, the sequence $(D(n\sqrt{d}))_{n=1}^{\infty}$ has infinitely many limit points.
Legendre found that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. Similarly, it is shown here that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of a composite $N$. Shanks' infrastructural method is then revisited, and some consequences of its application to factorization by means of the continued fraction expansion of $\sqrt N$ are derived.
For a monic polynomial $D(X)$ of even degree, express $\sqrt D$ as a Laurent series in $X^{-1}$; this yields a continued fraction expansion (similar to continued fractions of real numbers): \[\sqrt D=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\ddots}}},\quad a_i\text{ polynomials in }X.\] Such continued fractions were first considered by Abel in 1826, and later by Chebyshev. It turns out they are rarely periodic unless $D$ is defined over a finite field. Around 2001 van der Poorten studied non-periodic continued fractions of $\sqrt D$, with $D$ defined over the rationals, and simultaneously the…
The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$
本文证明了 $\sqrt{d}$ 连分数展开周期长度 $T(d)$ 前两阶矩的上界，从而改进了 $T(d)$ 大偏差的现有结果。
Let $\ell>3$ be a prime such that $\ell \equiv 3 \pmod{4}$ and $\mathbb{Q}(\sqrt{\ell})$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell})$ can be expressed as $h(-\ell) = (1/3)(b_1+b_2 + \cdots + b_m) - m$ where the $b_i$ are partial quotients in the `minus' continued fraction expansion $\sqrt{\ell} = [[b_0; \overline{b_1, b_2, \ldots, b_m}]]$. For an odd prime $p \neq \ell$, we prove an analogous formula using these $b_i$ which computes the sum of Iwasawa lambda invariants $λ_p(-\ell)+λ_p(-4)$ of $\mathbb{Q}(\sqrt{-\ell})$ and $\mathbb{…
Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF expansion. Hence it is a natural problem to determine all PICF expansions of irrational numbers. In this paper, we determine certain type PICF expansions of square roots of positive square-free integers. To obtain this result, it plays an important role to determine integer points on certain PCF varieties appeared in Brock-Elkies-Jordan. As an application of these …
On continued fraction partial quotients of square roots of primes
本文证明素数平方根连分数中特定部分商出现次数的有限性，给出周期长度可被4整除时部分商1的存在性，并建立周期长度上界及多项式分解，回答Miska和Ulas的若干问题。
We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of some of the continued fractions and show that polynomials arising from the generalizations are related to Chebyshev and Fibonacci polynomials. We then show that some of these…
Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$
本文研究Q(√127)中部分商≤4的有界连分数的周期与乘子序，证明相关精确公式，结合Hensley定理得到二次域允许无穷多此类连分数的数量下界。
The present paper proves that if for a power sum $α$ over $\ZZ$ the length of the period of the continued fraction for $\sqrt{α(n)}$ is constant for infinitely many even (resp. odd) $n$, then $\sqrt{α(n)}$ admits a functional continued fraction expansion for all even (resp. odd) $n$, except finitely many; in particular, for such $n$, the partial quotients can be expressed by power sums of the same kind.
Corrections are brought to an article of Friesen on continued fractions of a given period.
Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi--Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerato…
We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2, \dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ and generalize well-known families such as Chowla's $4n^2+1$, for which analogous results were recently proved by Dahl and Lamzouri.
Legendre discovered that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. In this vein, it was recently observed that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of composite $N$. It is proved here that these apparently fortuitous occurrences allow us to propose and apply a variation of Shanks' infrastructural method which significantly reduces the asymptotic computational burden with respect to currently used factoring techniques.
This paper is devoted to a detailed exposition of geometry of continued fractions. We pay particular interest to the case of quadratic irrationalities and use the technique described to prove a criterion for the continued fraction of a quadratic surd to have a symmetric period.
We study Möbius transformations (also known as linear fractional transformations) of quadratic numbers. We construct explicit upper and lower bounds on the period of the continued fraction expansion of a transformed number as a function of the period of the continued fraction expansion of the original number. We provide examples that show that the bound is sharp.
Anomalous Partial Quotients in the Continued Fraction of $\sqrt{ζ(3)-S_N}$
研究ζ(3)-S_N的连分数展开，通过Euler-Maclaurin展开及Pell恒等式确定特定指标下部分商规律，给出前五个部分商公式，揭示异常增长现象。
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical properties concerning to this subject.
Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficient conditions for pure periodicity of HCF. Then, we characterize badly approximable complex numbers in terms of HCF. Finally, we prove a slightly weaker complex analogue of a theorem by Y. Bugeaud on the transcendence of certain continued fractions.
A well known theorem of Lagrange states that the simple continued fraction of a real number $α$ is periodic if and only if $α$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.
We consider the geometric generalization of ordinary continued fraction to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely on these sails. This group transposes the faces. In this article, we present a method of constructing "approximate" fundamental domains of algebraic multidimensional continued fractions and an algorithm testing whether this domain is indeed fundamental or not. We give some polynomial estimates on number of the operations for the algorithm. In conclusion we p…
The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the $p$-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of $p$-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the pap…
We study the Kronecker symbol $\left(\frac st\right)$ for the sequence of the convergents $s/t$ of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.
An upper bound for the second moment of the length of the period of the continued fraction expansion for $\sqrt{d}$
本文改进了非平方数 $d$ 的 $\sqrt{d}$ 连分数周期长度二阶矩的上界，并修正余项估计与常数细节。
For a given sequence of positive integers we make an explicit construction of a reduced hyperbolic operator in SL(2,z) with the sequence as a period of a geometric continued fraction in the sense of Klein. Further we experimentally study an algorithm to construct a period for an arbitrary operator of SL(2,z) (the Gauss Reduction Theory).
We examine various properties of the continued fraction expansions of matrix eigenvector slopes of matrices from the SL(2, Z) group. We calculate the average period length, maximum period length, average period sum, maximum period sum and the distributions of 1s 2s and 3s in the periods versus the radius of the Ball within which the matrices are located. We also prove that the periods of continued fraction expansions from the real irrational roots of x2 + px + q = 0 are always palindromes.
In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in $\mathbb{Q}_p$ that have a periodic expansion with period of length four, solving an open problem left by Browkin.
Integer Factorization via Continued Fractions and Quadratic Forms
本研究提出一种基于SQUFOF方法相关理论的新型整数分解算法，分析表明其复杂度优于经典SQUFOF和CFRAC算法，且在特定条件下可达到多项式时间复杂度。
A continued fraction $v(τ)$ of Ramanujan is evaluated at certain arguments in the field $K = \mathbb{Q}(\sqrt{-d})$, with $-d \equiv 1$ (mod $8$), in which the ideal $(2) = \wp_2 \wp_2'$ is a product of two prime ideals. These values of $v(τ)$ are shown to generate the inertia field of $\wp_2$ or $\wp_2'$ in an extended ring class field over the field $K$. The conjugates over $\mathbb{Q}$ of these same values, together with $0, -1 \pm \sqrt{2}$, are shown to form the exact set of periodic points of a fixed algebraic function $\hat F(x)$, independent of $d$. These are analogues of similar resul…
Using continued fractions with prescribed period for universal quadratic forms
本文通过分析连分数周期与二次型判别式的同余条件，改进了实二次域上通用二次型的大秩结果。
In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the $(2,3,7)$-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem for the expansion which captures the fundamental relative units of quadratic extensions of $\mathbb{Q}(\cos(2π/7))$ with rank one relative unit groups. We also discuss the convergence of these continued fractions.
In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(τ)$, and are shown to be periodic points of an algebraic function.
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Per…
On Periodicity of Continued fractions with Partial Quotients in Quadratic Number Fields
针对部分商取自实二次域整数环的连分数，本文研究其最终周期性与收敛性，给出实四次无理数对应周期展开的充要条件及构造算法。
Periodic points in complex continued fractions by J. Hurwitz
本文通过自然扩张方法，明确刻画了 J. Hurwitz 复连分数算法中的纯周期点，并借鉴 Tanaka 的遍历理论结果，深化了对 Hurwitz 类 Lagrange 定理的理解。
Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the set \[\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34\}.\] We write explicitly all such polynomials for which the period $n$ occurs over $K$ but not over $\Q$ and $\displaystyle n\not\in\{ 13,15,17\}$. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period $13, 15$ or $17$ over $K$.
In this paper, we present some generalizations of Lagrange's theorem in the classical theory of continued fractions motivated by the geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension $F/F'$ of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to "expand" any basis of $F$ over $\mathbb{Q}$, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By developing the above argument adelically, we also obtain …
The present paper is in a sense a continuation of \cite{PLS}, it relies on the notation and some results. The problem tackled in both papers is the nature of the continued fraction expansion of $\sqrt[3]{2}$: are the partial quotients bounded or not. Numerical experiments suggest an even stronger result on the lines of Kuzmin statistics. Here we apply different sets of bases for the vector space $V$, where the adjunction ring $\ZZ\lbrack \sqrt[3]{2} \rbrack$ lives. And as a result we get a criterion for continued fraction convergents in terms of the coefficient vector from a lattice.
An effective criterion for periodicity of l-adic continued fractions
针对Ruban的l进连分数，本文给出有理数与二次无理数展开周期性的显式判据，补充有理数情形的定性结果。
It is known that the continued fraction expansion of a real number is periodic if and only if the number is a quadratic irrational. In an attempt to generalize this phenomenon to other settings, Jun-Ichi Tamura and Shin-Ichi Yasutomi have developed a new algorithm for multidimensional continued fractions (Algebraic Jacobi-Perron algorithm) that involves cubic irrationals, and proved periodicity in some cubic number fields, such as $\mathbb{Q}(\sqrt[3]{m^3+1})$ where $m\in\mathbb{Z}$, and $\mathbb{Q}(δ_m)$ where $δ_m$ is a root of $x^3-mx+1=0,\,\,m\in\mathbb{Z},\,\, m\geq3$ with the algorithm. …
Periodicity and Period-Length Bounds for Browkin $p$-Adic Continued Fractions
本文解决了Browkin $p$-进连分数关于二次无理数周期性的猜想，证明对任意正整数$t$，存在无穷多个整数平方根的展开周期长度为$2t$，并给出新的周期性充分条件与周期长度界，为Lagrange定理类似物研究提供工具。
Convergence, Finiteness and Periodicity of Several New Algorithms of p-adic Continued Fractions
本文提出若干改进的$p$进连分数展开算法，证明有理数部分商长度有界且小素数下二次无理数均可周期展开，并给出对任意素数构造此类算法的方法。
By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words, corresponding to continued fractions which are algebraic over F_2(t). This family includes a classical example already studied in different previous works by Y. Hu, G-N. Han, Y. Bugeaud and the author of this note.
Periodicity of general multidimensional continued fractions using repetend matrix form
本文研究一般多维连分数展开的周期性，通过刻画循环节对应矩阵的结构，证明若干向量在 Algebraic Jacobi–Perron 算法中具有最终周期展开，并给出纯周期展开的判据。
Real convergence and periodicity of $p$-adic continued fractions
本文研究了$p$-adic连续分数的实收敛与周期性关系，证明收敛是周期性的必要条件，并提出关于逆向的猜想，同时发展了非周期性的概率论论据。
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as $e$ or $π$ have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentiall…
We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real numbers sqrt(n) + n. It allows us to obtain that there exist infinitely many quadratic fields containing infinitely many continuous fraction expansions formed only by integers 1 and 2. We also prove that a conjecture of Zaremba implies a conjecture of McMullen, building periodic continuous fractions from continued fraction expansions of rational numbers.
p-Adically convergent loci in varieties arising from periodic continued fractions
该研究受\(\mathbb{Q}_p\)中元素连分数展开定义启发，通过考虑特定代数簇，研究具有\(\mathbb{Z}[1/p]\)中部分商的p进收敛周期连分数，刻画了这些簇的p进收敛轨迹的零维和一维情况。
$S$-units and period length of continued fractions of linear recursions
本文研究取值于实二次域的线性递推序列连分数周期长度的有界性问题，推广了 Corvaja 和 Zannier 先前对 $1$ 次线性递推情形的解决，并回应了 Mendès France 提出的关于 $α^n$ 连分数周期长度的猜想。
Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The numbers $a$ and $\sqrt b$ are uniquely determined by $x$. In general it is difficult to say what the influence of a certain digit of the repeating block on the appearance of $x$ is. We highlight a noteworthy exception from this rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit $c_n$ of the repeating block, the fractional part of $2a…
Transcendence criteria for multidimensional continued fractions
本文针对 Jacobi 引入的多维连分数建立超越性判据，证明某些 Liouville 型和拟周期多维连分数是超越的，并通过周期多维连分数产生的三次无理数高度上界完成拟周期情形的证明。
A new algorithm for p-adic continued fractions
本文提出一种改进的 p-adic 连分数算法，通过优化 Browkin 算法提升二次无理数展开的周期性，使其更接近拉格朗日定理的 p-adic 类似结论。
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a, a-a^2) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u,u') with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u,u') has a perio…
In this paper we show how to construct several infinite families of polynomials $D(\bar{x},k)$, such that $\sqrt{D(\bar{x},k)}$ has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter $k$. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in \cite{zannier} that the sequence of the…
Irrational Acceleration of a Continued Fraction of $π$
本文应用迭代Bauer-Muir加速方法，为$π$构造出收敛更快的无理系数Apéry-like连分数，并指出其矩阵表示与标准情形相同但维度增加，该构造同样适用于$\ln(2)$和$\sqrt[3]{2}$。
In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-ny^2 = \pm 1$. For each solution $x,y$ to Pell's equation we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $x+y\sqrt{n} \in \Z[\sqrt{n}]$. We further show that Ramanujan's dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.
Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\, q^2$. We show that the units $>1$ of the ring $\mathbb Z[\sqrt{Dq^2/Q}]$ are connected with certain convergents of $\sqrt{D/Q}$. Among these units, the units of $\mathbb Z[\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (…
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