Narrow Results

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s₀ such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s₀]) ⊂SL(2, ℂ). Let be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), and , where , μ ∈ ℤ[s₀]. The number μ can be recursively evaluated.

In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction [a₁, a₂,…,a_m], a_i ∈ ℕ, and the 2-bridge link of type p/q, where p/q is the rational number given by [a₁, a₂,…,a_m]. The known proof of this equivalence relies on the 2-fold cover of a link and the classification of the lens spaces. Our proof is elementary and combinatorial and follows the naïve approach of finding a set of movements to transform the rational link given by [a₁, a₂,…,a_m] into the 2-bridge link of type p/q.

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

Given a non-empty finite subset A of the natural numbers, let E_A denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, E_A is a Cantor set whose Hausdorff dimension dim(E_A) is between 0 and 1. It is shown that the set intersects [0,1/2] densely.

We then describe a method for accurately computing dimensions dim(E_A), and employ it to investigate numerically the way in which intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that is dense in [0,1].

In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim(E_{1,2}), improving on the one given in [18].

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, are given, for which the class groups H(K) are proved to contain cyclic subgroups of order n.

This paper gives explicit evaluations for two Ramanujan–Selberg continued fractions in terms of class invariants and singular moduli.

An exact formula for distribution of sequences of fractional parts is proved. This provides us with a possibility for computational study of their discrepancies by the means of integer-number programming.

In this paper a convergence theorem for continued fractions with negative parameters is proved. We define a transformation of a certain kind of continued fractions to the same kind of continued fractions. This transformation is obtained by multiple application of the Bauer–Muir transform and then using the limiting process. It is shown that a double application of this transformation is the identity transformation.

For x ∈ I, let [A₁(x), A₂(x), …] be the continued fraction expansions over the field of Laurent series, write L_n(x) ≔ max{deg A₁(x), deg A₂(x), …, deg A_n(x)}, which is called the largest degree of partial quotients. In this paper, we give an iterated logarithm type theorem for L_n(x), and by which, we get that for P-almost all x ∈ I, . Also the Hausdorff dimensions of the related exceptional sets are determined.

In this paper, we study 2-, 3-, 4-, 6- and 12-dissections of a continued fraction of order twelve. Our main result is that when the aforementioned continued fraction and its reciprocal are expanded as power series, the sign of the coefficients is periodic, with period 12.

Given x ∈ (0, 1), let [a₁(x), a₂(x), a₃(x),…] be the continued fraction expansion of x and be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {a_n(x), n ≥ 1} and proved that for any β > 0, the set

In this paper, we introduce the concept of F-perfect number, which is a positive integer n such that ∑_d|n,d<n d² = 3n. We prove that all the F-perfect numbers are of the form n = F_2k-1 F_2k+1, where both F_2k-1 and F_2k+1 are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.

Let p be a prime number. Let w₂ and denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies , with for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function takes any prescribed value in the interval (0, 1].

The Calkin–Wilf tree is a rooted infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. For every 2 × 2 matrix with nonnegative integral coordinates and nonzero determinant, it is possible to construct an analogous tree with this root. If the root is the identity matrix, then the tree consists of all matrices with determinant 1, and this tree possesses the basic properties of the Calkin–Wilf tree of positive rational numbers. The set of all matrices with nonnegative integral coordinates and nonzero determinant decomposes into a forest of rooted infinite binary trees.

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in $\mathbb{Z}[[x]]$ modulo $4$. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo $4$ are $2$-automatic.

Let $d$ be a positive square-free integer and ${𝜖}_d=(T_d+U_d\sqrt{d})/2>1$ be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$. The Ankeny–Artin–Chowla (AAC) conjecture asserts that $U_p\not\equiv 0$ (mod $p$) for primes $p\equiv 1$ (mod 4), which still remains unsolved. In this paper, sufficient conditions for $U_d<d$ have been given in terms of Yokoi’s invariants $n_d$ and $m_d$, and it has been shown that the AAC conjecture is true in some special cases.

For a given quadratic irrational $\alpha$, let us denote by $D(\alpha )$ the length of the periodic part of the continued fraction expansion of $\alpha$. We prove that for every positive integer $d$, which is not a perfect square, there are infinitely many even integers $k\ge 1$, for which the equality $D(n\sqrt{d})=k$ holds for infinitely many integers $n\ge 1$.

Let $a,b,c$ be any positive integers such that $c|ab$ and $d_i^{\pm }$ is a square free positive integer of the form $d_i^{\pm }=a^{2k}{b}^{2l}\pm i{c}^m$ where $k,l\ge m$ and $i=1,2.$ The main focus of this paper is to find the fundamental solution of the equation $x^2-d_i^{\pm }{y}^2=1,$ with the help of the continued fraction of $\sqrt{d_i^{\pm }}.$ We also obtain all the positive solutions of the equations $x^2-d_i^{\pm }{y}^2=\pm 1$ and $x^2-d_i^{\pm }{y}^2=\pm 4$ by means of the Fibonacci and Lucas sequences.

Furthermore, in this work, we derive some algebraic relations on the Pell form $F_{d_i^{\pm }}(x,y)=x^2-d_i^{\pm }{y}^2$ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $F_{{\mathrm{\Delta }}_{d_i^{\pm }}}(x,y)=1$ in terms of $d_i^{\pm }.$ We extend all the results of the papers [3, 10, 27, 37].