Explicit Algebraic Numbers all whose Integer Parts of Powers are Composite
We present several explicit classes of algebraic integers $\alpha>1$ such that $\lfloor \alpha^n\rfloor$ is composite for all but finitely many $n$.
math.NT↗
arXiv subjects
Dan Ismailescu
Publications and source records attributed to Dan Ismailescu.
Polynomially growing integer sequences all whose terms are composite
We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.
math.NT↗
A 5-chromatic same-distance graph in the hyperbolic plane
The chromatic number of the plane problem asks for the minimum number of colors so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the answer is 5, 6, or 7. Here we consider the problem in the context of the hyperbolic plane. We prove that there exists a distance $d\approx 1.375033509$ so that every 4-coloring of the hyperbolic plane contains two points distance $d$ apart, which are identically colored.
math.CO↗
The chromatic number of the Minkowski plane -- the regular polygon case
The Hadwiger-Nelson problem asks for the minimum number of colors, so that each point of the plane can be assigned a single color with the property that no two points unit-distance apart are identically colored. It is now known that the answer is $5$, $6$, or $7$, Here we consider the problem in the context of Minkowski planes, where the unit circle is a regular polygon with $8$, $10$, or $12$ vertices. We prove that in each of these cases, one also needs at least five colors.
math.CO↗
On the smallest area $n-1$-gon containing a convex $n$-gon
We prove that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$, and that every unit area convex hexagon is contained in a convex pentagon of area no greater than $7/6$. Both results are tight as the case of the regular pentagon (hexagon) shows. We conjecture that for every $n\ge 6$, every unit area convex $n$ - gon is contained in a $(n-1)$ - gon of area no greater than $1+\tan(2\pi/n)\sec(\pi/n)/n$.
math.MG↗
On the maximum area of inscribed polygons
Given a convex $n$-gon $P$ and a positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area convex $m$-gon contained in $P$. We are interested in the minimum value of $\Delta(Q)/\Delta(P)$, the ratio of the areas of these two polygons. More precisely, given positive integers $n$ and $m$, with $3 \le m \le n-1$, define \begin{equation*} f_n(m)=\min_{P\in \mathcal {P}_n} \max_{Q \subset P,|Q|=m} \frac{\Delta(Q)}{\Delta(P)} \end{equation*} where the maximum is taken over all $m$-gons contained in $P$, and the minimum is taken over $\mathcal{P}_n$, the entire class of convex $n$-gons. The values of $f_4(3)$, $f_5(4)$ and $f_6(3)$ are known. In this paper we compute the values of $f_5(3)$, $f_6(5)$ and $f_6(4)$. In addition, we prove that for all $n\ge 6$ we have \begin{equation*} \frac{4}{n}\cdot\sin^2\left(\frac{\pi}{n}\right)\le 1-f_n(n-1)\le \min\left(\frac{1}{n}, \frac{4}{n}\cdot\sin^2\left(\frac{2\pi}{n}\right)\right). \end{equation*} These bounds can be used to improve the known estimates for $f_n(m)$.
math.CO↗
Outdegree conditions forcing short cycles in digraphs
Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and H\"{a}ggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $\alpha(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le \alpha(m)$. This generalizes results of Shen who proved that $ch(3)\le 3-\sqrt{7}<0.35425$, and Liang and Xu who showed that $ch(4)< 0.28866$ and $ch(5)<0.24817$. We then slightly improve the above inequality by using the minimum feedback arc set approach initiated by Chudnovsky, Seymour, and Sullivan. This results in extensions of the findings of Hamburger, Haxell and Kostochka (in the case $m=3$), and Liang and Xu (in the case $m=4$).
math.CO↗
An improved upper bound for the size of the sphere of influence graph
Let $V$ be a set of $n$ points in the plane. For each $x\in V$, let $B_x$ be the closed circular disk centered at $x$ with radius equal to the distance from $x$ to its closest neighbor. The {\it closed sphere of influence graph} on $V$ is defined as the undirected graph where $x$ and $y$ are adjacent if and only if the $B_x$ and $B_y$ have nonempty intersection. It is known that every $n$-vertex closed sphere of influence graph has at most $cn$ edges, for some absolute positive constant $c$. The first result was obtained in 1985 by Avis and Horton who provided the value $c=29$. Their result was successively improved by several authors: Bateman and Erd\H{o}s (c=18), Michael and Quint (c=17.5), and Soss (c=15). In this paper we prove that one can take $c=14.5$.
math.CO↗
A $6$-chromatic two-distance graph in the plane
We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.
math.CO↗
Grid dissections of tangential quadrilaterals
For any integer $n\ge 2$, a square can be partitioned into $n^2$ smaller squares via a checkerboard-type dissection. Does there such a class-preserving grid dissection exist for some other types of quadrilaterals? For instance, is it true that a tangential quadrilateral can be partitioned into $n^2$ smaller tangential quadrilaterals using an $n\times n$ grid dissection? We prove that the answer is affirmative for every integer $n\ge 2$.
math.MG↗
The largest angle bisection procedure
The {\it largest angle bisection} procedure is the operation which partitions a given triangle, $T$, into two smaller triangles by constructing the angle bisector of the largest angle of $T$. Applying the procedure to each of these two triangles produces a partition of $T$ into four smaller triangles. Continuing in this manner, after $n$ iterations, the initial triangle is divided into $2^n$ small triangles. We prove that as $n$ approaches infinity, the diameters of all these $2^n$ triangles tend to $0$, the smallest angle of all these triangles is bounded away from $0$, and that, with the exception of $T$ being an isosceles right triangle, the number of dissimilar triangles is unbounded.
math.MG↗
On second order linear sequences of composite numbers
In this paper we present a new proof of the following 2010 result of Dubickas, Novikas, and Siurys: Let $(a,b)\in \mathbb{Z}^2$ and let $(x_n)_{n\ge 0}$ be the sequence defined by some initial values $x_0$ and $x_1$ and the second order linear recurrence \begin{equation*} x_{n+1}=ax_n+bx_{n-1} \end{equation*} for $n\ge 1$. Suppose that $b\neq 0$ and $(a,b)\neq (2,-1), (-2, -1)$. Then there exist two relatively prime positive integers $x_0$, $x_1$ such that $|x_n|$ is a composite integer for all $n\in \mathbb{N}$. The above theorem extends a result of Graham who solved the problem when $(a,b)=(1,1)$.
math.NT↗
On the polygon determined by the short diagonals of a convex polygon
Let $K$ be a convex pentagon in the plane and let $K_1$ be the pentagon bounded by the diagonals of $K$. It has been conjectured that the maximum of the ratio between the areas of $K_1$ and $K$ is reached when $K$ is an affine regular pentagon. In this paper we prove this conjecture. We also show that for polygons with at least six vertices the trivial answers are the best possible.
math.HO↗
Dense Packings with Nonparallel Cylinders
A \emph{cylinder packing} is a family of congruent infinite circular cylinders with mutually disjoint interiors in $3$-dimensional Euclidean space. The \emph{local density} of a cylinder packing is the ratio between the volume occupied by the cylinders within a given sphere and the volume of the entire sphere. The \emph{global density} of the cylinder packing is obtained by letting the radius of the sphere approach infinity. It is known that the greatest global density is obtained when all cylinders are parallel to each other and each cylinder is surrounded by exactly six others. In this case, the global density of the cylinder packing equals $π/\sqrt{12}= 0.90689\ldots$. The question is how large a density can a cylinder packing have if one imposes the restriction that \emph{no two cylinders are parallel}. In this paper we prove two results. First, we show that there exist cylinder packings with no two cylinders parallel to each other, whose local density is arbitrarily close to the local density of a packing with parallel cylinders. Second, we construct a cylinder packing with no two cylinders parallel to each other whose global density is $1/2$. This improves the results of K. Kuperberg, C. Graf and P. Paukowisch.
math.MG↗
The Hadwiger-Nelson problem with two forbidden distances
In 1950 Edward Nelson asked the following simple-sounding question: \emph{How many colors are needed to color the Euclidean plane $\mathbb{E}^2$ such that no two points distance $1$ apart are identically colored?} We say that $1$ is a \emph{forbidden} distance. For many years, we only knew that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey \cite{degrey} proved that at least five colors are necessary. In this paper we consider a related problem in which we require \emph{two} forbidden distances, $1$ and $d$. In other words, for a given positive number $d\neq 1$, how many colors are needed to color the plane such that no two points distance $1$ \underline{or} $d$ apart are assigned the same color? We find several values of $d$, for which the answer to the previous question is at least $5$. These results and graphs may be useful in constructing simpler $5$-chromatic unit distance graphs.
math.CO↗
The chromatic number of the plane is at least 5 - a new proof
We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points unit distance apart which are identically colored.
math.CO↗
On the Euclidean dimension of graphs
The Euclidean dimension a graph $G$ is defined to be the smallest integer $d$ such that the vertices of $G$ can be located in $\mathbb{R}^d$ in such a way that two vertices are unit distance apart if and only if they are adjacent in $G$. In this paper we determine the Euclidean dimension for twelve well known graphs. Five of these graphs, Dürer, Franklin, Desargues, Heawood and Tietze can be embedded in the plane, while the remaining graphs, Chvátal, Goldner-Harrary, Herschel, Fritsch, Grötzsch, Hoffman and Soifer have Euclidean dimension $3$. We also present explicit embeddings for all these graphs.
math.MG↗
On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$
The lower bound for the chromatic number of $\mathbb{R}^n$ is improved for $n = 6, 7, 10, 11, 12, 13 \mbox{ and } 14$.
math.CO↗