On pp-adic valuations of colored pp-ary partitions

Let $m\in\N_{\geq 2}$ and for given $k\in\N_{+}$ consider the sequence $(A_{m,k}(n))_{n\in\N}$ defined by the power series expansion $$\prod_{n=0}^{\infty}\frac{1}{\left(1-x^{m^{n}}\right)^{k}}=\sum_{n=0}^{\infty}A_{m,k}(n)x^{n}.$$ The number $A_{m,k}(n)$ counts the number of representations of $n$ as sums of powers of $m$, where each summand has one among $k$ colors. In this note we prove that for each $p\in\mathbb{P}_{\geq 3}$ and $s\in\N_{+}$, the $p$-adic valuation of the number $A_{p,(p-1)(p^s-1)}(n)$ is equal to 1 for $n\geq p^s$. We also obtain some results concerning the behaviour of the sequence $(\nu_{p}(A_{p,(p-1)(up^s-1)}(n)))_{n\in\N}$ for fixed $u\in\{2,\ldots,p-1\}$ and $p\geq 3$. Our results generalize the earlier findings obtained for $p=2$ by Gawron, Miska and the first author.Comment: 10 pages, to appear in Monatshefte f\"{u}r Mathemati

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer $n$ has the Lehmer property if $\phi(n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$\left\{\frac{g^{n}-1}{g-1}\ \bigg|\ n,g\in\mathbb{N},\ \nu_{2}(g)+\nu_{2}(g+1)\leq L\ \right\},$$ where $\nu_{2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real number.Comment: 4 page

Special economic zones (SEZ) in the crisis caused by the COVID-19 pandemic

The aim of the publication is to analyse and evaluate the functioning of special economic zones in Poland, focusing in particular on the capital invested in the zones and on the creation of new jobs. This publication addresses the subject of special economic zones in Poland and the conditions for their functioning. The article focuses on the structure of capital invested in the zones, the number of workers and industry specialisation. Special attention was paid to issues related to the functioning of SEZs in Poland in the conditions of the global crisis caused by the COVID-19 pandemic

Real quadratic fields with a universal form of given rank have density zero

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.Comment: 18 pages, minor change

Efektywność wykorzystania środków pomocowych WPR współfinansujących projekty inwestycyjne na przykładzie małych gospodarstw rolnych województwa małopolskiego

Celem artykułu jest ocena efektywności wykorzystania środków pomocowych WPR współfinansujących projekty inwestycyjne, które realizowane były w latach 2004-2015 przez małe gospodarstwa rolne z województwa małopolskiego. Badania wykazały, że gospodarstwa najczęściej wykorzystywały środki pomocowe na realizację inwestycji polegających na uzupełnianiu i unowocześniana swojego parku maszynowego oraz budowie lub modernizacji obiektów gospodarskich, co doprowadziło do powiększenia zasobów rzeczowych tych gospodarstw, a także unowocześnienia prowadzonej w nich produkcji. Badania wykazały stosunkowo niską efektywność wykorzystania środków pomocowych WPR współfinansujących projekty inwestycyjne. Ponadto stwierdzono istnienie istotnej statystycznie, dodatniej zależności między wielkością gospodarstwa, a efektywnością wykorzystania środków

Effect of European funds on the development of agricultural farms in Poland

The paper strives to present the level of diversification of agricultural development in Poland resulting from the uneven utilization of European funds allocated to this aim. Development of agriculture is determined significantly by the scale of investment in this sector. Therefore, this article analyses the measure “Modernization of agricultural farms” of RDP 2007-2013 and the state of the utilization of its measures by agricultural farms in the individual regions of Poland. The degree of activity and efficiency of beneficiaries in acquisition of European funds were determined in the paper and an attempt was made at identification of factors diversifying i