arXiv subjects

Piotr Miska

Publications and source records attributed to Piotr Miska.

At least 19 recordsLinked to original sources

arXiv2025

On the arithmetic of multidimensional continued fractions

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the M\"obius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms.

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arXiv2024

Binary sequences meet the Fibonacci sequence

We introduce a new family of meta-Fibonacci sequences $(f(n))_{n\in\mathbb{N}}$, governed by the recurrence relation $$f(n)=af(n-u_{n}-1)+bf(n-u_{n}-2),$$ where $\mathbf{u}=(u_{n})_{n\in \mathbb{N}}$ is a sequence with values $0,1$. Our study focuses on the properties of the sequence of quotients $h(n) = f(n+1)/f(n)$ and its set of values $\mathcal{V}(f)=\{h(n): n \in \mathbb{N}\}$ for various $\mathbf{u}$. We give a sufficient condition for finiteness of $\mathcal{V}(f)$ and automaticity of $(h(n))_{n \in \mathbb{N}}$, which holds in particular when $\mathbf{u}$ is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software \texttt{Walnut}. On the other hand, we prove that the set $\cal{V}(f)$ is infinite for other special binary sequences $\mathbf{u}$, and obtain a trichotomy in its topological type when $\mathbf{u}$ is eventually periodic.

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arXiv2024

Connections between certain numbers related to derangements and $r$-permutations

For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \sigma(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } \sigma(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\} \end{align*} and \begin{align*} \cal{D}_{r,u,m}(n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x<y\leq r} \ x \textrm{ and } y \textrm{ are in disjoint cycles of } \sigma \textrm{ and } \sigma(z)=z \textrm{ for exactly } u \textrm{ elements } 1\leq z\leq r, \textrm{ and } \sigma(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\}, \end{align*} where $\mathcal{S}_{n}$ denotes the set of all the permutations of $\{1,\ldots ,n\}$. In this paper we study connections between the sets $\mathcal{D}(r,u,m,n)$, $\mathcal{D}_{r,u,m}(n)$, and the sets of (some classes of) $r$-derangements. We rely mostly on counting arguments.

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arXiv2024

On practical sets and $A$-practical numbers

Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$-practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\right\}$ can be written as a sum of distinct divisors of $n$ that belong to $A$. Denote the set of $A$-practical numbers as $\text{Pr}(A)$. The aim of the paper is to explore the properties of the sets $\text{Pr}(A)$ (the form of the elements, cardinality) as $A$ varies over the power set of $\mathbb{N}$. We are also interested in the set-theoretic and dynamic properties of the mapping $\mathcal{PR}:\mathcal{P}(\mathbb{N})\ni A\mapsto\text{Pr}(A)\in\mathcal{P}(\mathbb{N})$.

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arXiv2023

On general approach to Bessenrodt-Ono type inequalities and log-concavity property

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt-Ono type inequalities and log-concavity property. In this note we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function $F$ of at most exponential growth satisfying the condition $F(\mathbb{N})\subset \mathbb{R}_{+}$, we have $F(a)F(b)>F(a+b)$ for sufficiently large positive integers $a, b$. Moreover, we show that if the sequence $(F(n))_{n\geq n_{0}}$ is log-concave and $\limsup_{n\rightarrow +\infty}F(n+n_{0})/F(n)<F(n_{0})$, then $F$ satisfies the Bessenrodt-Ono type inequality.

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arXiv2023

On alternative definition of Lucas atoms and their $p$-adic valuations

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.

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arXiv2023

On continued fraction partial quotients of square roots of primes

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime number and $D=p$ or $2p$ is such that the length of the period of continued fraction expansion of $\sqrt{D}$ is divisible by $4$, then $1$ appears as a partial quotient in the continued fraction of $\sqrt{D}$. Furthermore, we give an upper bound for the period length of continued fraction expansion of $\sqrt{D}$, where $D$ is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas.

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arXiv2022

Characteristics of distributions of sets and their $(R)$- and $(N)$-denseness

Let $0\leq q\leq1$ and $\mathbb{N}$ denotes the set of all positive integers. In this paper we will deal with it too the family $\mathcal{U}(x^q)$ of all regularly distributed set $X \subset \mathbb{N}$ whose ratio block sequence is asymptotically distributed with distribution function $g(x) = x^q;\ x \in(0,1]$, and we will show that the regular distributed set, regular sequences, regular variation at infinity are equivalent notations. In this paper also we discuss the relation ship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion of sequence and exponent of convergence.

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arXiv2022

On Frobenius problem with restrictions on common divisors of coefficients

Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1μ_1+a_2μ_2+\ldots+a_sμ_m$, where positive integers $μ_1,μ_2,\ldots,μ_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $μ_1,μ_2,\ldots,μ_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be written as a sum of positive integers $μ_1,μ_2,\ldots,μ_n$ with no common divisor being $m$-th power of a positive integer greater than $1$ but each $s-1$ of the values of $μ_1,μ_2,\ldots,μ_s$ have a common divisor being $m$-th power of a positive integer greater than $1$.

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arXiv2022

On Waring numbers of henselian rings

Let $n>1$ be a positive integer. Let $R$ be a henselian local ring with residue field $k$ of $n$th level $s_n(k)$. We give some upper and lower bounds for the $n$th Waring number $w_n(R)$ in terms of $w_n(k)$ and $s_n(k)$. In large number of cases we are able to compute $w_n(R)$. Similar results for the $n$th Waring number of the total ring of fractions of $R$ are obtained. We then provide applications. In particular we compute $w_n(\mathbb{Z}_p)$ and $w_n(\mathbb{Q}_p)$ for $n\in\{3,4,5\}$ and any prime $p$.

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arXiv2022

On the Diophantine equation $σ_{2}(\overline{X}_{n})=σ_{n}(\overline{X}_{n})$

In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of $σ_{2}(\overline{X}_{n})$ and $σ_{n}(\overline{X}_{n})$ together with the biggest value of the variable $x_{n}$ appearing in the solution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss the set of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.

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arXiv2022

Binomial coefficients, roots of unity and powers of prime numbers

Let $t\in\mathbb{N}_+$ be given. In this article we are interested in characterizing those $d\in\mathbb{N}_+$ such that the congruence $$\frac{1}{t}\sum_{s=0}^{t-1}{n+dζ_t^s\choose d-1}\equiv {n\choose d-1}\pmod{d}$$ is true for each $n\in\mathbb{Z}$. In particular, assuming that $d$ has a prime divisor greater than $t$, we show that the above congruence holds for each $n\in\mathbb{Z}$ if and only if $d=p^r$, where $p$ is a prime number greater than $t$ and $r\in\{1,\ldots ,t\}$.

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arXiv2022

$p$-Adic quotient sets: diagonal forms

For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic denseness of quotients of nonzero values attained by diagonal forms of degree $n\geq 3$, where $\gcd(n,p(p-1))=1$.

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arXiv2022

On distribution of subsequences of primes having prime indices with respect to the $(R)$-denseness and convergence exponent

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define subsequences $(p^{(k)}_n)_{n=1}^{+\infty}$ of the sequence $(p_n)_{n=1}^{+\infty}$ in the following way: let $p_n^{(1)}=p_n$ and $p_n^{(k+1)}=p_{p_n^{(k)}}$. In this paper we study and describe some interesting properties of the sets $\mathbb{P}_k=\{p_1^{(k)}<p_2^{(k)}<\dots<p_n^{(k)}<\dots\}$, $\mathbb{P}_n^{\mathrm{T}}=\{p_n^{(1)}<p_n^{(2)}<\dots<p_n^{(k)}<\dots\}$ and $\text{Diag}\mathbb{P}=\{p^{(1)}_1<p^{(2)}_2<\dots <p^{(k)}_k<\dots\}$ and their elements, for $k,n\in\mathbb{N}$. Especially, we check whether these sets have dense sets of ratios in $\mathbb{R}_+$. Moreover, we compute their exponents of convergence and asymptotics of their counting functions.

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arXiv2021

Stirling number and periodic points

We introduce the notion of almost realizability, an arithmetic generalization of realizability for integer sequences, which is the property of counting periodic points for some map. We characterize the intersection between the set of Stirling sequences (of both the first and the second kind) and the set of almost realizable sequences.

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arXiv2020

A note on $p$-adic denseness of quotients of values of quadratic forms

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.

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arXiv2020

On two conjectures regarding generalized sequence of derangements

The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in \cite{miska}. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.

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arXiv2019

On some properties of the number of permutations being products of pairwise disjoint $d$-cycles

Let $d\geq 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_d(n))_{n\in\N}$, where $H_{d}(n)$ is the number of permutations in $S_{n}$ being products of pairwise disjoint cycles of a fixed length $d$. In particular we deal with periodicity modulo a given positive integer, behaviour of the $p$-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study they properties. Among many results we obtain qualitative description of the $p$-adic valuation of the number $H_{d}(n)$ extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida.

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