arXiv subjects

Maciej Ulas

Publications and source records attributed to Maciej Ulas.

At least 19 recordsLinked to original sources

arXiv2026

Hankel determinants of weighted binary sums of digits

Let $s_\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\mathbf{w}=(w_j)_{j\geq 0}$. We investigate Hankel determinants $\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i+j)]_{0\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\mathcal{H}_\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\mathcal{H}_\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \mathcal{H}_\mathbf{w}(\lceil 2^{k+2}/3\rceil)= (-1)^{\frac{(k+2)(k+3)}{2}}(k+1). $$ This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization $w_j=t^j$, where the determinants become polynomials in $t$, and investigate their vanishing. For $t=2ζ$, where $ζ$ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to $\mathcal{H}_\mathbf{w}(n)$, we consider Hankel determinants associated with the first difference of $s_{\mathbf{w}}$, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.

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arXiv2026

There are infinitely many Hilbert cubes of dimension 3 in the set of squares

A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set $\cal{S}=\{n^2:\;n\in\mathbb{N}\}$ of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if $H(a_{0}; a_{1}, \ldots, a_{d})\subset \cal{S}\cap [0, N]$, then $d\leq 7 \log\log N$ for all sufficiently large values of $N$. Here we prove that there exist at least $\gg N^{1/8}$ Hilbert cubes $H(a_{0}; a_{1}, a_{2}, a_{3})$ with $a_{0}, a_{1}, a_{2}, a_{3}\in [0,N]$ in the set of squares. Moreover, we prove that for each $i, j\in\{0, 1, 2, 3\}$ with $i<j$, the set $$ \left\{\frac{a_{i}}{a_{j}}:\;H(a_{0}; a_{1}, a_{2}, a_{3})\subset S\right\} $$ is dense in the set of positive real numbers (in the Euclidean topology).

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arXiv2025

On primitive integer solutions of the Diophantine equation $x^3\pm y^3=a^k\pm b^k$

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive integer solutions. For $k=5, 7$ and all choices of the signs we computed all co-prime positive integer solutions $(x, y, a, b)$ satisfying the condition $\op{max}\{a, b\}\leq 50000$.

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arXiv2025

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

Construction of diagonal quintic threefolds with infinitely many rational points

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over $\Q$ each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples $B=(B_{0}, B_{1}, B_{2}, B_{3})$ of co-prime integers such that for a suitable chosen integer $b$ (depending on $B$), the equation $B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b$ has infinitely many positive integer solutions.

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arXiv2024

Signs behaviour of sums of weighted numbers of compositions

Let $A$ be a subset of positive integers. For a given positive integer $n$ and $0\leq i\leq n$ let $c_{A}(i,n)$ denotes the number of $A$-compositions of $n$ with exactly $i$ parts. In this note we investigate the sign behaviour of the sequence $(S_{A,k}(n))_{n\in\N}$, where $S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{k}i^{k}c_{A}(i,n)$. We prove that for a broad class of subsets $A$, the number $(-1)^{n}S_{A,k}(n)$ is non-negative for all sufficiently large $n$. Moreover, we show that there is $A\subset \N_{+}$ such that the sign behaviour of $S_{A,k}(n)$ is not periodic.

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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

Geometric progressions in the sets of values of rational functions

Let $a, Q\in\Q$ be given and consider the set $\cal{G}(a, Q)=\{aQ^{i}:\;i\in\N\}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$. Let $f\in\Q(x, y)$ and $\cal{V}_{f}$ be the set of finite values of $f$. We consider the problem of existence of $a, Q\in\Q$ such that $\cal{G}(a, Q)\subset\cal{V}_{f}$. In the first part of the paper we describe several classes of rational function for which our problem has a positive solution. In particular, if $f(x,y)=\frac{f_{1}(x,y)}{f_{2}(x,y)}$, where $f_{1}, f_{2}\in\Z[x,y]$ are homogenous forms of degrees $d_{1}, d_{2}$ and $|d_{1}-d_{2}|=1$, we prove that $\cal{G}(a, Q)\subset \cal{V}_{f}$ if and only if there are $u, v\in\Q$ such that $a=f(u, v)$. In the second, experimental, part of the paper we study the stated problem for the rational function $f(x, y)=(y^2-x^3)/x$. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.

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arXiv2022

Values of binary partition function represented by a sum of three squares

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that $b_{m}(n)$ can not be represented as a sum of three squares and it is equal to $1/12$ for $k=1, 2$ and $1/6$ for $k\geq 3$. In particular, for $m=1$ the equation $b_{1}(n)=x^2+y^2+z^2$ has a solution in integers if and only if $n$ is not of the form $2^{2k+2}(8s+2t_{s}+3)+i$ for $i=0, 1$ and $k, s$ are non-negative integers, and where $t_{n}$ is the $n$th term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in $n$ of the equation $b_{2^k-1}(n)=x^2+y^2+z^2$.

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arXiv2022

Solutions of certain meta-Fibonacci recurrences

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.

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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

Signs behaviour of sums of weighted numbers of partitions

Let $A$ be a subset of positive integers. By $A$-partition of $n$ we understand the representation of $n$ as a sum of elements from the set $A$. For given $i, n\in\N$, by $c_{A}(i,n)$ we denote the number of $A$-partitions of $n$ with exactly $i$ parts. In the paper we obtain several result concerning sign behaviour of the sequence $S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{i}i^{k}c_{A}(i,n)$, where $k\in\N$ is fixed. In particular, we prove that for a broad class $\cal{A}$ of subsets of $\N_{+}$ we have that for each $A\in \cal{A}$ we have $(-1)^{n}S_{A,k}(n)\geq 0$ for each $n, k\in\N$.

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arXiv2021

A Fibonacci type sequence with Prouhet-Thue-Morse coefficients

Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb N}$, where $h_0 = 0, h_1 = $1 and for $n \ge 2$ we define $h_n$ recursively as follows:$ h_n = t_n h_{n-1} + h_{n-2}$. We prove several results concerning arithmetic properties of the sequence $(h_n )_{n\in \mathbb N}$. In particular, we prove non-vanishing of $h_n$ for $n \ge 5$, automaticity of the sequence $(h_n \pmod m)_{n\in \mathbb N}$ for each m, and other results.

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arXiv2021

Divisibility and Arithmetic Properties of a Class of Sparse Polynomials

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider rational roots. This leads to the study of an infinite class of integer sequences which have interesting properties and satisfy linear recurrence relations.

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arXiv2021

Equal values of certain partition functions via Diophantine equations

Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite sets.

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arXiv2021

Diophantine problems related to cyclic cubic and quartic fields

We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.

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arXiv2021

Some observations and speculations on partitions into $d$-th powers

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_{d}(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\geq 2$. Besides results concerning the asymptotic behavior of $p_{d}(n)$ a little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers. The second part of the paper has experimental nature and contains questions and conjectures concerning arithmetic behavior of the sequence $(p_{d}(n))_{n\in\N}$. They based on our computations of $p_{d}(n)$ for $n\leq 10^5$ in case of $d=2$, and $n\leq 10^{6}$ for $d=3, 4, 5$.

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arXiv2020

Some Properties of a Class of Sparse Polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.

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