In this paper we establish the distribution of prime numbers in a given arithmetic progression $p \equiv l \hspace{4.44443pt}(\@mod \; k)$ for which $ap + b$ is squarefree.
We consider positional numeration system with negative base −β, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when β is a quadratic Pisot number. We study a class of roots β>1 of polynomials x2−mx−n, m≥n≥1, and show that in this case the set Fin(−β) of finite (−β)-expansions is closed under addition, although it is not closed under subtraction. A particular example is β=τ=12(1+5–√), the golden ratio. For such β, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of (−τ)-integers coincides on the positive half-line with the set of (τ2)-integers.
The celebration of The Wallachian Year held in 1925 significantly influenced the development of folklore movement in the ethnographic region of Moravian Wallachia. This event inspired Arnošt Kubeša to promote traditional folk music and dances as part of his teaching career. For this purpose he founded with his students the first Wallachian Circle in the mid-1930s, he organised its public performances (as well as the first foreign tour of this kind of an ensemble) and continued to found other circles in the end. After his involuntary retirement from the education system due to his "political and ideological unreliability" and his withdrawal from the leading positions in folklore movement associations, Arnošt Kubeša started a new career as a museum employee. This study refers to his activities which contributed to the development of folklore in its second existence.