Related papers: Closed formulae for certain Fermat-Pell equations

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…

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007),…

Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers $ d $ such that the Pell…

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…

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…

A continued fraction $v(\tau)$ 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…

Let $K$ be a totally real number field and consider a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat…

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…

For a set of positive integers $D$, a $k$-term $D$-diffsequence is a sequence of positive integers $a_1<a_2<\cdots<a_k$ such that $a_i-a_{i-1}\in D$ for $i=2,3,\cdots,k$. For $k\in\mathbb{Z}^+$ and $D\subset \mathbb{Z}^+$, we define…

Given a prime $\ell\geq 3$ and a positive integer $k \le \ell-2$, one can define a matrix $D_{k,\ell}$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian ${\mathrm Jac}({\mathcal…

We consider the Lorentz force equation $$ \frac{d}{dt}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = q \left(E(t,x) + \dot x \times B(t,x)\right), \qquad x \in \mathbb{R}^3, $$ in the physically relevant case of a singular…

Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…

The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the…

Let $z$ be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums $S(p_k,q_k)$ belonging to convergents $p_k/q_k$ of the {\em regular} continued fraction expansion of $z$ with that of Dedekind sums $S(s_j/t_j)$…

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic…

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show…

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of…

Let $\mathbb{F}_q$ be a finite field with $q=p^t$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ over $\mathbb{F}_q$. A classic well-konwn result from Weil yields a…

We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\geq 1$, we improve the error term in the partial sums of the number of…