Quadratic Irrational Research Articles

On radiuses of convergence of q-metallic numbers and related q-rational numbers

The q-rational numbers and the q-irrational numbers are introduced by S. Morier-Genoud and V. Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of Fibonacci numbers and Pell numbers which are introduced by S. Morier-Genoud and V. Ovsienko. We give an estimation of radiuses of convergence of them, and we solve on conjecture of the lower bound expected which is introduced by L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov for the metallic numbers and its convergence rational sequence. In addition, we obtain a relationship between the radius of convergence of the [n,n,...,n,...]_q and [n,n,...,n]_q in the case of n=3 and n=4.

Mobius Group Generated by Two Elements of Order 2, 4, and Reduced Quadratic Irrational Numbers

The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the set Q ′ ′ m = η / s : η ∈ Q ∗ m , s = 1 , 2 under the action of group H = < x ′ , y ′ : x ′ 2 = y ′ 4 = 1 > , where m is square free integer and Q ∗ m = a ′ + m / c ′ , a ′ , a ′ 2 − m / c ′ c ′ = 1 , c ′ ≠ 0 . We discuss the type and reduced cardinality of the orbit Q ′ ′ p . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime p . Further, we classify that for a reduced number r whether − r , r ¯ , − r ¯ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.

On extreme values for the Sudler product of quadratic irrationals

For a real number $\alpha $ and a natural number $N$, the Sudler product is defined by $P_N(\alpha ) = \prod _{r=1}^{N} 2 \lvert \sin (\pi r\alpha )\rvert $. Denoting by $F_n$ the $n$th Fibonacci number and by $\phi $ the Golden Ratio, we show that for $F

On Hermite’s problem, Jacobi–Perron type algorithms, and Dirichlet groups

A well-known result of Lagrange (1770) characterises quadratic irrationalities as those real numbers that can be written as periodic continued fractions. Hermite asked in 1848 if there exists some way to write cubic irrationalities periodically. To ap

Systems of joint Thue polynomials for quadratic irrationalities

Systems of joint Thue polynomials for quadratic irrationalities

Periodic Representations and Approximations of p-adic Numbers Via Continued Fractions

Continued fractions can be introduced in the field of p-adic numbers Q p , however currently there is not a standard algorithm as in R . Indeed, it is not known how to construct p-adic continued fractions that give periodic representations for all quadratic irrationals and provide good p-adic approximations. In this article, we introduce a novel algorithm which terminates in a finite number of steps when processes rational numbers. Moreover, we study when it provides particular periodic representations of period 2 and pre-period 1 for quadratic irrationals. We also provide some numerical experiments regarding periodic representations and p-adic approximations of quadratic irrationals, comparing the performances with Browkin’s algorithm presented in [6], which is one of the most classical and interesting algorithm for continued fractions in Q p .

A quaternionic construction of p-adic singular moduli

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which \({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture.

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call E-reduced, and respectively B-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.

Q-Deformations in the modular group and of the real quadratic irrational numbers

We develop further the theory of q-deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of matrices of the modular group PSL(2,Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the q-deformed matrices are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q-deformed quadratic irrationals.

Periodic representations for quadratic irrationals in the field of 𝑝-adic numbers

Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.

Class invariants by Siegel resolvents and the modularity of their Galois traces

The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level.

Peter Chew Theorem and Application

The Objective of Peter Chew Theorem is to make it easier and faster to solve the problem of quadratic roots, by converting any value of the Quadratic Surds ( √(a+b√(c) ) ) into the sum or difference of two real numbers. Peter Chew Theorem can also convert the square root of a complex number into a complex number (the sum or difference between the real part and the imaginary part), because the square root of the complex number is also the quadratic surd ( √(a+bi) =√(a+b√(-1)) ). In addition, Peter Chew's Theorem can also convert Quadratic Surds ( √(a+b√(c ) )) into the sum or difference of two complex number (√z +√(z ̅ ) ). Technical tools have had a significant impact on advanced mathematics teaching and mathematics learning. Symbolab raised $1.2 million to introduce its unique math search engine to smartphones and tablets. However, today's online calculator only contains the knowledge that has been explained in the book, but the current method cannot or is difficult to solve some Quadratic Surds problems, which makes the online calculator unable to solve the Quadratic Surds problems. This will cause students to reduce interest and hinder the promotion of the use of technical tools. In order to solve the mention problems, my research is to create a new discovery (Peter Chew theorem) for the topic of Quadratic Surds, so that all problems can be easily solved, and then apply Peter Chew’s theorem to a new calculator (PCET calculator), Allow the PCET calculator to solve any problem in the topic of Quadratic Surds, which can make the PCET calculator effectively help mathematics teaching, especially in the future when similar COVID-19 problems arise. Videos related to this article can be obtained through the following links: https://youtu.be/9m7mc0UTsSw.

Gaussian behavior of quadratic irrationals

We study the probabilistic behavior of the continued fraction expansion of a quadratic irrational number, when weighted by some “additive” cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dyn

Singular moduli for real quadratic fields: A rigid analytic approach

A rigid meromorphic cocycle is a class in the first cohomology of the discrete group Γ : = SL 2 ( Z [ 1 / p ] ) with values in the multiplicative group of nonzero rigid meromorphic functions on the p -adic upper half-plane H p : = P 1 ( C p ) − P 1 ( Q p ) . Such a class can be evaluated at the real quadratic irrationalities in H p , which are referred to as “RM points.” Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of Γ -orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the values of modular functions on SL 2 ( Z ) H at complex multiplication (CM) points: in particular, they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p -adic accuracy leads to convincing numerical evidence for the algebraicity and factorization of the resulting singular moduli for real quadratic fields.

Symmetries of a two-dimensional continued fraction

We describe the symmetry group of a multidimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: Dirichlet symmetries, which correspond to the multiplication by units of the respective extension of , and so-called palindromic symmetries. The main result is a criterion for a two-dimensional continued fraction to have palindromic symmetries, which is analogous to the well-known criterion for the continued fraction of a quadratic irrationality to have a symmetric period.

A More Suitable Definition of Quadratic Surds

Background:The properties of the groupPGL(2,C) on the Upper Poincar´e Half Plane have been analyzed.Objective:In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem.Methods:The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found.Results:A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)).Conclusion:The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori.

Symmetries of Icosahedral group and classification of G-circuits of length six

Objectives: To determine the exact number of equivalence classes of G-circuits of length q≥2:Methods/Statistical Analysis: To classify G-orbits of Q(√m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t. Applications/Improvements: We employ Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example. Keywords: Rotational symmetries of icosahedral group; partition function; equivalence classes of G-circuits; reduced quadratic irrational numbers

On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field ofpp-adic numbersQp\mathbb {Q}_p. MCFs have also been recently introduced inQp\mathbb {Q}_p, including, in particular, app-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of thepp-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finitepp-adic MCF representsQ\mathbb Q-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF havepp-adic valuations equal to−1-1. Finally, we show that a periodic MCF of dimensionmmconverges to an algebraic irrational of degree less than or equal tom+1m+1; for the casem=2m=2, we are able to give some more detailed results.

A Generalization of the Secant Zeta Function as a Lambert Series

Recently, Lalín, Rodrigue, and Rogers have studied the secant zeta function and its convergence. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. They also gave modular transformation properties of the secant zeta function. In this paper, we generalized secant zeta function as a Lambert series and proved a result for the Lambert series, from which the main result of Lalín et al. follows as a corollary, using the theory of generalized Dedekind eta-function, developed by Lewittes, Berndt, and Arakawa.

Properties of circuits in coset diagrams by modular group

Background/Objectives: Graham Higman gave the idea of coset diagrams for the action of modular group PSL(2;Z) on real quadratic irrationals. These special types of graphical figures are composed of closed paths known as Circuits. These circuits can be classified into certain types of even length with respect to the number of inside\\outside triangles. This study is to discuss different properties of reduced numbers in coset diagrams of the type (p;q). Methods: In this study, we have investigated different properties of type (p; q) using reduced quadratic irrationals and continued fractions. We have categorized reduced numbers in accordance with their position in the real line. Distance between two ambiguous numbers and reduced numbers is introduced in this article which will help the reader to understand the structural significance of reduced numbers in a circuit. We have explored different conditions under which certain reduced numbers have the same circuit. Moreover, continued fractions have been used to assist the foundation laid by modular group action and different general results have been derived in this context. Findings: It was possible to define new notions of equivalent, cyclically equivalent and similar circuits using partitions of n and discuss various properties of reduced numbers included in coset diagrams of circuits with length up to four.