Related papers: On a problem of Peth\H{o}

Let $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

We show how to determine the $k$-th bit of Chaitin's algorithmically random real number $\Omega$ by solving $k$ instances of the halting problem. From this we then reduce the problem of determining the $k$-th bit of $\Omega$ to determining…

We study periodic representations in number systems with an algebraic base $\beta$ (not a rational integer). We show that if $\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such…

Let $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and $\beta\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there…

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that…

For any given positive integer $m$ we construct certain totally positive algebraic integers $\alpha$ of a real bi-quadratic field $K$ and obtain some necessary conditions for which $m\alpha$ can not be represented as sum of integral…

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric…

For $\alpha>1$, set $\beta=1/(\alpha-1)$. We show that, for every $1<\alpha<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^\alpha}\rfloor - \lfloor{m^\alpha}\rfloor$ is equal to…

We give a characterization of symplectic quadratic Lie algebras that their Lie algebra of inner derivations has an invertible derivation. A family of symplectic quadratic Lie algebras is introduced to illustrate this situation. Finally, we…

Let $P(x):=a_d x^d+\cdots+a_0\in\mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\geq 2$. Let $(x_n)$ be a sequence of integers satisfying \begin{equation*} x_{n+1}=P(x_n)\mbox{for all}\quad n=0,1,2\ldots,\quad\mbox{and} \quad…

The purpose of this article is to give the solutions of the inverse problem for Pellian equations. For any rational number $0< a/b < 1$, the fundamental discriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$ are given…

We study rational numbers with purely periodic R\'enyi $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\gamma(\beta)=1$, i.e., that all rational…

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic''…

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

In this paper, we present some numerical applications for the equation $x^2+ax+b=0$, where $a, b$ are two quaternionic elements in $\mathbb{H}(\alpha,\beta)$. Based on well-known solving methods, we have developed a new numerical algorithm…

Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not…

Schinzel and W\'ojcik have shown that if $\alpha, \beta$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(\alpha)=\mathrm{ord}_p(\beta)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in…

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd…