The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call $$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$ its $q$-ary expansion. We call a function $f$ strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e., $$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$ Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0\>0$, ${\beta }_0\>\cdots \>{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial. ...

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 $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set $${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$ Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series...

$\alpha$-normality and $\beta$-normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff $\beta$-normal non-normal space and an example of a Hausdorff $\alpha$-normal non-regular space.

Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $ℱ\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus ℱ$ is $\sigma$-porous and such that for each $A\in ℱ$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.