Triples Of Integers Research Articles

The Golden Circle and the Aureate Circle

We define golden triples to be triples of integers satisfying a particular quadratic equation. The first two elements of each triple are consecutive terms of a Fibonacci-like sequence. We show that each golden triple can be represented by a rational point on a particular circle, which we call the golden circle. We also generate a related circle called the aureate circle.

Averaged forms of two conjectures of Erdős and Pomerance, and their applications

Let P+(n) denote the largest prime factor of an integer n. In this paper, we study the asymptotics of sums∑p⩽xP+(p+h)⩽y1and∑n⩽x,P+(n)⩽y1P+(n+h)⩽y21 for xε⩽y,y1,y2⩽x. Our results imply that two conjectures concerning the asymptotics for the above two sums, which were considered by Erdős and Pomerance, hold on average. As an application of the first sum above, we can deduce that there are infinitely many primes p⩽x such that P+(p+h)>p1−ε for all but o((log⁡x)c0) values of shifts h, where c0 is a small computable constant. As applications of the second sum above, we consider the averaged version of Erdős–Turán's conjecture and the equation a+b=c. In particular, we show that there exists at least a positive density (>17.505%) of triples of integers {a,b,c} such that a+b=c and c≪P+(a)P+(c). This upper bound is stronger than c≪(∏p|abcp)1+ε in the abc-conjecture.

GENERATING DIOPHANTINE TRIPLES RELATING TO FIGURATE NUMBERS WITH THOUGHT-PROVOKING PROPERTY

In this article, the process for constructing integer triples consisting of some figurate numbers specifically Star number, Centered square number and Centered hexagonal number where the arithmetic mean of any pair of the elements remains a perfect square is scrutinized. Also, Python program for conforming all the triples sustaining the desired constraint for numerical values is established.

A group of Pythagorean triples using the inradius

Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

Local wave numbers on an n-spheres

We consider Eigen-functions of the Laplace–Beltrami Operator on n-Spheres and characterize them in terms of their local plane wave behavior. We estimate the local spectrum of wave numbers by approximating the Spherical harmonics in the locally flat neighborhood around a point on the Spheres. These local wave numbers are shown to obey an interesting Pythagorean type relation. Based on this relation, we propose a question whether there are integer triples for 2-spheres and their generalization to n-spheres. We apply the local spectrum to define quantities such as phase velocity and group velocity on a sphere and outline the relevance of the analysis for the case fields on de Sitter space.

The length of an s-increasing sequence of r-tuples

AbstractWe prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

Groups and monoids of Pythagorean triples connected to conics

Abstract We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

Only finitely many Tribonacci Diophantine triples exist

Abstract Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

The gauge action, DG Lie algebras and identities for Bernoulli numbers

Abstract In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers ( a , b , c ) ${(a,b,c)}$ with a + b + c = n - 1 ${a+b+c=n-1}$ , n ≥ 4 ${n\geq 4}$ . These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

A combinatorial coordinate system for the body-centered cubic grid

A new compact combinatorial coordinate system is presented to address not only the voxels (truncated octahedra) of the body-centered cubic grid but also the lower-dimensional cells – faces, edges and vertices. Almost all integer triples address some cell in the grid, and incidence and adjacency relations between cells can easily be captured by the coordinate values. The new coordinate system can effectively and efficiently be applied in topological shape analysis and modeling, image processing and computer graphics applications.

Legendrian𝜃-graphs

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar theta-graph. We define the transverse push-off of a Legendrian graph and we determine its self linking number for Legendrian theta-graphs. In the case of topologically planar theta-graphs, we prove that the topological type of the transverse push-off is that of a pretzel link.

Diophantine approximation with sign constraints

Let $\alpha$ and $\beta$ be real numbers such that $1$, $\alpha$ and $\beta$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max\{|x_1|,|x_2|\}^{-2}$. In 1976, W. M. Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma\cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb{Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max\{|x_1|,|x_2|\}^\gamma$ is arbitrarily small. Although Schmidt later conjectured that $\gamma$ can be replaced by any number smaller than $2$, N. Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$. In this paper, we present a construction showing that the result of Schmidt is in fact optimal.

A note on Diophantine systems involving three symmetric polynomials

Let X ¯ n = ( x 1 , … , x n ) and σ i ( X ¯ n ) = ∑ x k 1 … x k i be i -th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers a , b , c such that for each 1 ⩽ i ⩽ n the system of Diophantine equations σ i ( X ¯ 2 n ) = a , σ 2 n − i ( X ¯ 2 n ) = b , σ 2 n ( X ¯ 2 n ) = c has infinitely many rational solutions. This result extends the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each k there are at least k n -tuples of integers with the same sum of i -th powers for i = 1 , 2 , 3 . Similar result is proved for i = 1 , 2 , 4 and i = − 1 , 1 , 2 .

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.

90.34 On triples of integers having the same sum and the same product

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Some new triples of integers and associated triangles

Some new triples of integers and associated triangles

Pythagorean triples and sums of triangular numbers

Each primitive Pythagorean triple of positive integers leads, with the aid of an algebraic identity, to a family of triples of integers. Each triple in this family provides three triangular numbers, one for each component, such that the sum of two of them is equal to the third one.

Noether-Lefschetz, space curves and mathematical instantons

There are three areas of study of geometric entities living in the three dimensional projective space over the complex numbers which have proved to be strictly connected: the classification of space curves, surfaces and vector bundles. Despite all the production in these areas, several key unsolved problems still challenge us. After the work of many classical and modern authors (see [H1], [H2] for a bibliography) the problem of classifying space curves has been divided into two main parts: 1) (the so called Halphen problem) Find all the triples of integers (n, g, s) such that there exists a smooth irreducible curve C ⊂ IP 3 of degree n and genus g lying on a surface of degree s but not on any surface of lower degree; 2) Study the properties of the Hilbert scheme Hn,g of smooth irreducible curves in IP , namely its irreducible components, dimension, singular points. Let G(n, s) be the maximum genus of a curve of degree n not contained in a surface of degree less than s. Then part 1 above falls in turn into three ranges for n and s:

Simultaneous Pellian equations

In this paper we describe a method for finding integer solutions of simultaneous Pellian equations, that is, integer triples (x,y,z) satisfying equations of the formwhere the coefficientsa,b,c,d,fare integers and we assume thata,c, andacare not square.

A Note on Alternating Series in Several Dimensions

In the course of a study of chemical lattice [1] the authors considered such as (1.1) ∑′ (- 1) n+m+k ( n2 + m2 + k2) -1/2, the summation being over all non-zero integer triples. Such sums occur naturally in the study of crystal potentials. For example, (1.1) is meant to measure the potential at the origin of an infinite cubic crystal with unit Coulomb charges at each integer lattice point. As such the sum is considered to represent an electrochemical constant (Madelung's constant) for sodium chloride. An excellent account of such lattice can be found in Glasser and Zucker's recent survey [3].