We investigate properties of coset topologies on commutative domains with an identity, in particular, the 𝓢-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster...

In the two dimensional real vector space ${ℝ}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^2$. We construct points, where...

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider the Golomb topology ${\tau }_G$ on the set $\mathbb{N}$ of natural numbers, as well as the Kirch topology ${\tau }_K$ on $\mathbb{N}$. Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in $(\mathbb{N},{\tau }_G)$. We also show that $(\mathbb{N},{\tau }_G)$ and $(\mathbb{N},{\tau }_K)$ are aposyndetic. Our results...

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit.

We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1,u_2,u_3,\cdots$ in base $b$ such that the numbers $u_0{b}^n+u_1{b}^{n-1}+\cdots +u_{n-1}b+u_n,$ where $n=0,1,2,\cdots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \{2,3,4,6\},$ and that no unavoidable set exists in base $b=5.$ Now, we prove...

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $\left(\right(xy/q\left)\right),(0\le x,y\&lt;q)$, where $\left(\right(u\left)\right)=u-\left[u\right]-1/2$ denotes the “centered” fractional part of $x$. These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$-functions at $s=1$.