Visible points on multidimensional modular hyperbolas

For integers q?1, s?3 and a with gcd(a,q)=1 and a real U?0, we obtain an asymptotic formula for the number of integer points (u₁,…,us)∈s[1,U] on the s-dimensional modular hyperbola with the additional property gcd(u₁,…,us)=1. Such points have a geometric interpretation as points on the modular hyperbola which are “visible” from the origin. This formula complements earlier results of the first author for the case s=2 and a=1. Moreover, we prove stronger results for smaller U on “average” over all a. The proofs are based on the Burgess bound for short character sums.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

The least common multiple of a sequence of products of linear polynomials

Let f(x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log?lcm?(f(1),…,f(n))～An as n→∞, where A is a constant depending on?f.     

On the number of distances between the coordinates of points on modular hyperbolas

For a prime p>2, an integer a with gcd(a,p)=1 and 1?X,Y<p we give an asymptotic formula for the number of different Euclidean distances |x−y| defined by the points on the modular hyperbola . Furthermore, in the case X=Y=p−1 we determine the exact number of different distances.     

The least common multiple of random sets in polynomial rings over finite fields

The Ramanujan Journal - In this paper we investigate the distribution of degrees of the least common multiples of random subsets of monic polynomials of degree n in $${\mathbb {F}}_q[t]$$ . We...     

The permutation of integers with small least common multiple of two subsequent terms

Erd?s, Freud and Hegyvári [1] constructed a permutation a ₁,a ₂,… of positive integers with \([a_{i}, a_{i+1}]< i\exp \left\{c\sqrt{\log i}\log\log i\,\right\}\) for an absolute constant c>0 and all i≧3. In this note, we construct a permutation of all positive integers such that for any ε>0 there exists an i ₀ with \([a_{i}, a_{i+1}]\allowbreak < i\exp \left\{\left(2\sqrt{2}+\varepsilon\right) \sqrt{\log i\log\log i}\,\right\}\) for all i≧i ₀.     

The lattice points of ann-dimensional tetrahedron

We show that the number of orderedm-tuples of points on the integer lattice, inside or on then-dimensional tetrahedron bounded by the hyperplanesX ₁=0,X ₂=0, ...,X _n=0 andw ₁ X ₁+w ₂ X _n+...+w _n X_n=X, with the property that, for eachj, no more thank such points have non-zerojth ordinate, is asymptotically

Fibonacci lattice points

Odd indexed Fibonacci numbers F2n+1 can be written as sums of two squares a ²+b ². In this paper, we study the distribution of the lattice points (a,b) on the circles of radius \(\sqrt{F_{2n}+1}\).     

The number of lattice points on a convex curve

If C is a strictly convex plane curve of length l, it has been known for a long time that the number of integer lattice points on C is $\text{O(l}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{)}$ and the exponent is best possible. In this paper, it is shown that the exponent can be decreased by imposing suitable smoothness conditions on C; in particular, if C has a continuous third derivative with a sensible bound, the best possible value of the exponent lies between $\text{3}\text{5}$ and $\text{1}\text{3}$ inclusive.     

A Diophantine problem of Frobenius in terms of the least common multiple

The Diophantine Problem of Frobenius is to find a formula for the least integer not representable as a nonnegative linear form of positive integers. A reduction formula for the Diophantine Problem of Frobenius is presented. The formula can be applied whenever there are common divisors of the coefficients except for the whole set of them. The reduction formula is expressed in terms of the least common multiple of the coefficients. For some classes of coefficients this formula gives an exact answer for the problem of Frobenius, and these classes are fully characterized in the paper.     

Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms

Let l and m be two integers with l > m ≥ 0, and let a and b be integers with a ≥ 1 and a + b ≥ 1. In this paper, we prove that log lcm _{mn < i ≤ ln} {ai + b} = An + o(n), where A is a constant depending on l, m and a.     

A remark on Feuerbach hyperbolas

Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J Geom Graph 14:1–14, 2010; Kimberling et al. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2012; Moses and Kimberling J Geom Graph 13:15–24, 2009; Moses and Kimberling Forum Geom 11:83–93, 2011; Odehnal Elem Math 61:74–80, 2006; Odehnal Forum Geom 10:35–40, 2010; Odehnal J Geom Graph 15: 45–67, 2011). The aim of this paper is to present a new approach to the construction of points on the Feuerbach hyperbola. Surprisingly, these points can be obtained as centers of perspectivity of a triangle ABC and a certain one-parametric set of triangles A′B′C′. The presented construction is based on partitions of the triangle’s sides and—in a way—dual to the construction of points on the Kiepert hyperbola. It can also be generalized to spherical triangles. The proofs are based on an affine property of triangles, which amazingly can also be used for the proof of the spherical theorem.     

Convex lattice polygons with all lattice points visible

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.     

Improvements of lower bounds for the least common multiple of finite arithmetic progressions

Let and be positive integers such that . Let for . We prove that if . This improves the lower bound of obtained previously by Farhi, Hong and Feng.

     

Greatest common divisors and least common multiples of graphs

A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG ₁ andG ₂ ifG is a graph of maximum size for whichG|G ₁ andG|G ₂, while a graphH without isolated vertices is a least common multiple ofG ₁ andG ₂ ifH is a graph of minimum size for whichG ₁|H andG ₂|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG ₁ andG ₂ to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG ₁,G ₂ of graphs are determined, including when one ofG ₁ andG ₂ is a cycle of even length and the other is a star.G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060