Proof without words: Periodic continued fractions

Периодичность и квазипериодичность функциональных непрерывных дробей в гиперэллиптическом поле $$L = \mathbb{Q}(x)(\sqrt{f})$$ имеет более сложную природу, чем периодичность числовых непрерывных дробей элементов квадратичных полей. Известно, что периодичность непрерывной дроби элемента $$\sqrt{f}/h^{g+1}$$, построенной по нормированию, связанному с многочленом h первой степени, эквивалентна наличию нетривиальных S-единиц в поле L рода g и эквивалентна наличию нетривиального кручения в группе классов дивизоров. В данной статье найден точный промежуток значений $$s \in \mathbb{Z}$$ таких, что элементы $$\sqrt{f}/h^s$$ имеют периодическое разложение в непрерывную дробь, где $$f \in \mathbb{Q}[x]$$ - свободный от квадратов многочлен четной степени. Для многочленов f нечетной степени проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ рассмотрена в статье [5], причем там доказано, что длина квазипериода не превосходит степени фундаментальной S-единицы поля L. Проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ для многочленов f четной степени является более сложной. Это подчеркивается найденным нами примером многочлена f степени 4, для которого соответствующие непрерывные дроби имеют аномально большую длину периода. Ранее в статье [5] также были найдены примеры непрерывных дробей элементов гиперэллиптического поля L с длиной квазипериода значительно превосходившей степень фундаментальной S-единицы поля L.

This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.

Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.

defined over the field of rational numbers Q is normal. They do this by noting that, over a field of characteristic 3, F (X) equals 1/ √ 1 + 1/X. The latter has a periodic continued fraction and, as a consequence, they are able to explicitly verify that F (X) is normal. They then use an “automatic” proof to show that the Laurent series of F (X), defined over Q, is normal. Here we shall give a different proof of the latter result (Theorem 4, below) and show, more generally, that if a formal Laurent series Γ (X) defined over Q is defined and normal (mod p) for any prime p, then it is also normal over Q. We shall also consider the continued fraction of the more general surd 1/ √ 1 + u/X + v/X2 over any field not of characteristic 2. Put δ = u/4−v and note that √ 1 + u/X + v/X2 = 1 + u/(2X) when δ = 0. Hence, when δ = 0, the continued fraction expansion is trivial and is not normal. Define

In the classical case, the connection between the periodicity of the continued fraction of $$\sqrt f $$ and the existence of a fundamental unit of the corresponding hyperelliptic field $$\mathcal{L} = K(x)(\sqrt f )$$ , where K is a field of characteristic different from 2, has long been known. For the element $$\sqrt f $$ , the period length of the continued fraction in $$K((1{\text{/}}x))$$ can be trivially estimated from above by the doubled degree of the fundamental unit. Much more complicated and interesting is the problem of estimating (from above) the period length of other elements of $$\mathcal{L}$$ that have a periodic continued fraction. Among these elements, those of the form $$\sqrt f {\text{/}}{{x}^{s}}$$ , $$s \in \mathbb{Z}$$ , play a key role. For such elements, the period length can be many times greater than the double degree of the fundamental unit. In this article, we find upper bounds for the period length of key elements of hyperelliptic fields $$\mathcal{L}$$ over number fields K. An example is found that demonstrates the sharpness of the proven upper bounds.

This chapter offers an overview of words and quadratic numbers, and in particular ordering the conjugates of a Christoffel word. Within this topic the reader learns that the reversal of a Christoffel word is a conjugate, and that the lower and upper Christoffel words of the same slope are the smallest and the largest in their conjugation class. The chapter discusses computation in terms ofMarkoff numbers of the quadratic real number which has a periodic continued fraction with periodic pattern equal to a Christoffel word written on the alphabet 11, 22. It also reviews computation of theMarkoff supremum of a periodic biinfinite sequence, and of theLagrange number of a periodic sequence, both having a periodic pattern as above.

Periodic continued fractions and elliptic curves

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

This Demonstration provides a way to instantly write a periodic general continued fraction representation of square roots. This can be used as a quick way to easily approximate the numerical value of square roots. (As opposed to a general continued fraction a simple continued fraction has all of its numerators equal to 1.)The proof of this representation is quite simple; let . Then so and simp;;

Periodic expansion of modules and its relation to units

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

Let K ( a n / b n ) K({a_n}/{b_n}) be a limit periodic continued fraction of elliptic type; i.e., a n → a {a_n} \to a and b n → b {b_n} \to b , where a / ( b + w ) a/(b + w) is an elliptic linear fractional transformation of w . We show that if ∑ | a n − a | > ∞ \sum {|{a_n} - a| > \infty } and ∑ | b n − b | > ∞ \sum {|{b_n} - b| > \infty } , then K ( a n / b n ) K({a_n}/{b_n}) diverges. This generalizes the well-known Stern-Stolz Theorem. The Gauss continued fraction (related to hypergeometric functions) is used as an example. We also give an example where a n − a = O ( n − 1 ) {a_n} - a = \mathcal {O}({n^{ - 1}}) and b n = b = 1 {b_n} = b = 1 and K ( a n / b n ) K({a_n}/{b_n}) converges. The divergence result is also generalized further.

Fast computation of continued fractions