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Piotr Miska

Publications and source records attributed to Piotr Miska.

26 records · Page 2Linked to original sources
arXiv2019

On consecutive 1's in continued fractions expansions of square roots of prime numbers

In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption of Hypothesis H of Schinzel.

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arXiv2018

$p$-adic denseness of members of partitions of $\mathbb{N}$ and their ratio sets

The ratio set of a set of positive integers $A$ is defined as $R(A) := \{a / b : a, b \in A\}$. The study of the denseness of $R(A)$ in the set of positive real numbers is a classical topic and, more recently, the denseness in the set of $p$-adic numbers $\mathbb{Q}_p$ has also been investigated. Let $A_1, \ldots, A_k$ be a partition of $\mathbb{N}$ into $k$ sets. We prove that for all prime numbers $p$ but at most $\lfloor \log_2 k \rfloor$ exceptions at least one of $R(A_1), \ldots, R(A_k)$ is dense in $\mathbb{Q}_p$. Moreover, we show that for all prime numbers $p$ but at most $k - 1$ exceptions at least one of $A_1, \ldots, A_k$ is dense in $\mathbb{Z}_p$. Both these results are optimal in the sense that there exist partitions $A_1, \ldots, A_k$ having exactly $\lfloor \log_2 k \rfloor$, respectively $k - 1$, exceptional prime numbers; and we give explicit constructions for them. Furthermore, as a corollary, we answer negatively a question raised by Garcia et al.

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arXiv2018

On the p-adic denseness of the quotient set of a polynomial image

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in $\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots + x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a question posed in [Garcia et al., $p$-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question.

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arXiv2018

The $r$-derangement numbers

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, $p$-adic valuations, and diophantine problems.

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arXiv2017

Arithmetic properties of coefficients of power series expansion of $\prod_{n=0}^{\infty}\left(1-x^{2^{n}}\right)^{t}$ (with an Appendix by Andrzej Schinzel)

Let $F(x)=\prod_{n=0}^{\infty}(1-x^{2^{n}})$ be the generating function for the Prouhet-Thue-Morse sequence $((-1)^{s_{2}(n)})_{n\in\N}$. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function $$ F_{t}(x)=F(x)^{t}=\sum_{n=0}^{\infty}f_{n}(t)x^{n}. $$ For $t\in\N_{+}$ the sequence $(f_{n}(t))_{n\in\N}$ is the Cauchy convolution of $t$ copies of the Prouhet-Thue-Morse sequence. For $t\in\Z_{<0}$ the $n$-th term of the sequence $(f_{n}(t))_{n\in\N}$ counts the number of representations of the number $n$ as a sum of powers of 2 where each summand can have one among $-t$ colors. Among other things, we present a characterization of the solutions of the equations $f_{n}(2^k)=0$, where $k\in\N$, and $f_{n}(3)=0$. Next, we present the exact value of the 2-adic valuation of the number $f_{n}(1-2^{m})$ - a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.

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arXiv2015

Arithmetic Properties of the Sequence of Derangements and its Generalizations

The sequence of derangements is given by the formula $D_0 = 1, D_n = nD_{n-1} + (-1)^n, n>0$. It is a classical object appearing in combinatorics and number theory. In this paper we consider two classes of sequences: first class is given by the formulae $a_0 = h_1(0), a_n = f(n)a_{n-1} + h_1(n)h_2(n)^n, n>0$, where $f,h_1,h_2 \in\mathbb{Z}[X]$, and the second one is defined by $a_n = \sum_{j=0}^n \frac{n!}{j!} h(n)^j, n\in\mathbb{N}$, where $h\in\mathbb{Z}[X]$. Both classes are a generalization of the sequence of derangements. We study such arithmetic properties of these sequences as: periodicity modulo $d$, where $d\in\mathbb{N}_+$, $p$-adic valuations, asymptotics, boundedness, periodicity, recurrence relations and prime divisors. Particularly we focus on the properties of the sequence of derangements and use them to establish arithmetic properties of the sequences of even and odd derangements.

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arXiv2014

A note on $p$-adic valuations of the Schenker sums

A prime number $p$ is called a Schenker prime if there exists such $n\in\mathbb{N}_+$ that $p\nmid n$ and $p\mid a_n$, where $a_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\in\mathbb{N}_+$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ has exactly one solution modulo $5^k$. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if $37\nmid n$ then $v_{37}(a_n)\leq 1$, what means that the second conjecture stated by the mentioned authors is not true.

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