A space $X$ is splittable over a space $Y$ (or splits over $Y$) if for every $A\subset X$ there exists a continuous map $f:X\to Y$ with $f^{-1}fA=A$. We prove that any $n$-dimensional polyhedron splits over ${𝐑}^{2n}$ but not necessarily over ${𝐑}^{2n-2}$. It is established that if a metrizable compact $X$ splits over ${𝐑}^n$, then $dimX\le n$. An example of $n$-dimensional compact space which does not split over ${𝐑}^{2n}$ is given.

For $\varnothing \ne M\subseteq {\omega }^{*}$, we say that $X$ is quasi $M$-compact, if for every $f:\omega \to X$ there is $p\in M$ such that $\overline{f}\left(p\right)\in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi ${\omega }^{*}$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in ${\omega }^{*}$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M\subseteq {\omega }^{*}$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah...

Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D,p)$, and also prove that if the equation $U^2-D{V}^2=-1$ has integer solutions $(U,V)$, the least solution $(u_1,v_1)$ of the equation $u^2-p{v}^2=1$ satisfies $p\nmid v_1$, and $D>C\left(p\right)$, where $C\left(p\right)$ is an effectively computable...