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Topological aspects of infinitude of primes in arithmetic progressions

František Marko, Štefan Porubský (2015)

Colloquium Mathematicae

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...

Totally Brown subsets of the Golomb space and the Kirch space

José del Carmen Alberto-Domínguez, Gerardo Acosta, Gerardo Delgadillo-Piñón (2022)

Commentationes Mathematicae Universitatis Carolinae

A topological space $X$ is totally Brown if for each $n \in ℕ ∖ {1}$ and every nonempty open subsets $U_{1}, U_{2}, ..., U_{n}$ of $X$ we have ${cl}_{X} (U_{1}) \cap {cl}_{X} (U_{2}) \cap \dots \cap {cl}_{X} (U_{n}) \neq \emptyset$ . Totally Brown spaces are connected. In this paper we consider the Golomb topology $τ_{G}$ on the set $ℕ$ of natural numbers, as well as the Kirch topology $τ_{K}$ on $ℕ$ . 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 $(ℕ, τ_{G})$ . We also show that $(ℕ, τ_{G})$ and $(ℕ, τ_{K})$ are aposyndetic. Our results...

Displaying 141 – 151 of 151

Topological aspects of infinitude of primes in arithmetic progressions

Totally Brown subsets of the Golomb space and the Kirch space

Towards a Katona type proof for the 2-intersecting Erdős-Ko-Rado theorem.

Translated geometric progressions and covering systems

Über die kleinsten quadratfreien Zahlen in arithmetischen Progressionen.

Van der Waerden's theorem and avoidability in words.

Variations on a theme of Sierpiński.

Vector-covering systems with a single triple of equal moduli

Well-distributed 2-colorings of integers relative to long arithmetic progressions

Zur Kongruenzklassen-Verteilung in Folgen von kleinen Zahlen, insbesondere von Primzahldifferenzen.

$λ$ -primitive roots

Currently displaying 141 – 151 of 151