In this note we prove that the equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1=q^n+1$, $q\ge 2,n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2·{10}^{17}$.

Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes...

Let $D$ be a Carathéodory domain. For $1\le p\le \infty$, let $L^p\left(D\right)$ be the class of all functions $f$ holomorphic in $D$ such that ${\parallel f\parallel }_{D,p}=[\frac{1}{A}\int {\int }_D^{}{\left|f\left(z\right)\right|}^p{dxdy]}^{1/p}<\infty$, where $A$ is the area of $D$. For $f\in L^p\left(D\right)$, set $$E_n^p\left(f\right)=\underset{t\in {\pi }_n}{inf}{\parallel f-t\parallel }_{D,p};$$ ${\pi }_n$ consists of all polynomials of degree at most $n$. In this paper we study the growth of an entire function in terms of approximation error in $L^p$-norm on $D$.

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ(*)$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$\begin{array}{c}r>0,q\ge 0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right),\\ \underset{\left|d\right|\to \infty }{lim}{\int }_{x-d}^x\frac{\mathrm{d}t}{r\left(t\right)}·{\int }_{x-d}^{x}q\left(t\right)\mathrm{d}t=\infty .\end{array}$$ In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p\left(ℝ\right),$ $p\in (1,\infty ).$ In this criterion, we use values of some auxiliary implicit functions in the coefficients $r$ and $q$ of equation ($*$). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function $f\left(x\right)$ for $x\in (a,b)$ through a function $g\left(x\right)$ is sharp by order if $c^{-1}\left|g\left(x\right)\right|\le \left|f\left(x\right)\right|\le c\left|g\left(x\right)\right|,$ ...

We prove: If $A\left(n\right)$ and $G\left(n\right)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA\left(n\right)/G\left(n\right)-(n-1)A(n-1)/G(n-1)$ $(n\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty$.