Arithmetic Progressions Research Articles

Sharp weak‐type estimate for maximal operators associated to Cartesian families under an arithmetic condition

AbstractGiven a set of integers , we consider the smallest family invariant by translations that contains the rectangles for any and where for integer. We prove that if contains arbitrary large arithmetic progression, then the maximal operator associated to the family satisfies a weak‐type estimate of the form but does not satisfy a weak‐type estimate of the form for any nonnegative convex increasing function such that as tends to infinity.

Any Two-Coloring of the Plane Contains Monochromatic 3-Term Arithmetic Progressions

Any Two-Coloring of the Plane Contains Monochromatic 3-Term Arithmetic Progressions

Algorithms for representing positive odd integers as the sum of arithmetic progressions

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

Menyelasaikan Masalah Kehidupan Sehari-hari dalam Konsep Barisan dan Deret Aritmatika

This research aims to solve everyday life problems using the concept of arithmetic sequences and series. The type of research used is qualitative research. The type of data used is secondary data. Data collection method uses library research. The method for this study uses previous literature studies, books and journals. An arithmetic sequence is a number that has a certain pattern and a fixed difference. while an arithmetic series is the sum of all the numbers in the arithmetic series. The concept of arithmetic sequences and series can be applied in life as a solution to solve everyday life problems.

Ethnomathematics Study at The Tomb of Sunan Bonang Tuban

Ethnomathematics is mathematics inherent in the form of local wisdom. This study aims to describe the ethnomathematics at the Tomb of Sunan Bonang Tuban. These Data were obtained through observation, interviews, and documentation. The Data is analyzed by reducing the data, presenting the data, and drawing conclusions. To obtain the validity of the data, a credibility test through triangulation, a transferability test through a detailed description, as well as a dependability test and confirmability test by auditing. The results of ethnomathematics studies on the Tomb of Sunan Bonang Tuban in the aspect of counting there <em>candra sengkala</em> on gate I. In the aspect of locating obtained patterns of sequential arrangement of pages, the arrangement of pages and gates that have a repeating pattern, the arrangement of Pendopo <em>rante</em> sirap and Sunan Bonang Tomb cungkup sirap that form an arithmetic sequence, the arrangement of cemaric plates on the <em>rana</em> that form a pattern of numbers, and the arrangement of other tombs are parallel. In designing aspects obtained page I and III rectangular, trapezoidal page II, shingle Pavilion <em>rante</em> and Sunan Bonang Tomb <em>cungkup</em>-shaped combination of rectangular pyramid and truncated rectangular pyramid.

MAGIC SQUARES OF PERFECT SQUARES AND PELL NUMBERS

Order three magic squares of distinct squared integers are studied. We show that such a magic square is not possible if the smallest entry is the square of a prime number, or unity. A method for generating all arithmetic progressions of three squared integers whose smallest term is the square of a prime or unity is presented via a set of linear transformation matrices involving the Pell numbers.

A note on the distribution of $$d_3(n)$$ in arithmetic progressions

A note on the distribution of $$d_3(n)$$ in arithmetic progressions

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Effective results for polynomial values of (alternating) power sums of arithmetic progressions

Effective results for polynomial values of (alternating) power sums of arithmetic progressions

On the set of values of a positive ternary quadratic form

It is shown that the set of values of an integer positive definite quadratic form in three variables contains an infinite arithmetic progression.

Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей

A matrix model of subsequence of natural numbers with a multiplicative basis from the first prime numbers is proposed. The matrix model is a square matrix, where vector columns are arithmetic progressions with difference and number of progressions equal to product of base elements. By crossing out arithmetic progressions with first terms which are multiples of base elements, we obtain a symmetric sparse matrix that contains all the prime numbers of subsequence of natural numbers, except for the base ones, which increases the density of prime numbers in sparse matrices. Sparse matrices are not formed clearly, only the vector of first terms of arithmetic progressions is formed. Properties of sparse matrices have been proven. Formulas that speed up the calculation of compound numbers in arithmetic progressions have been derived, the structures of vector elements of first terms of arithmetic progressions have been determined, the connectivity of symmetric parts of sparse matrices has been investigated. With the expansion of the base the number of pairs of elements with the difference equal to the power of two («twins», «fours», etc.) increases in the vector of first terms. This is a necessary condition for the existence of constants for which linear equations of two variables can have an infinite set of solutions in prime numbers. The irregularity of distribution of prime numbers in subsequences of natural numbers is related to the structure of elements of the vector of first terms. An algorithm for finding prime numbers on segments of large dimensions with a parallel calculation process has been built. The proposed algorithm is binary to algorithms for sifting subsequences of natural numbers by prime divisors. In these algorithms, it is not possible to parallelize the calculation process, since screening procedure requires storing the numerical information from the preceding steps (vector model of array processing). The binary algorithm calculates compound numbers in each pair of arithmetic progressions with symmetric first terms simultaneously, using only vector of the first terms of arithmetic progressions, which makes possible processing of large-dimensional arrays

Mathematical principles of mechanical movements

The article proves that kinematic and dynamic theories in classical mechanics by I. Newton, considered a mathematical model of body displacement, are based on the laws of arithmetic progression (or linear function), trigonometry and quadratic function (equation). It is shown that formulas, equations of motion and graphs of dependencies in these theories are mathematical mechanisms that implement the relationship between uniformly varying mechanical quantities and quantities that implement (modify) changes. It is shown that the transition of a uniform change of one value to an accelerated (progressive) change occurs according to the scheme of transition of a linear function into a quadratic equation. It is shown that the formula of an arithmetic progression (linear function) is created by moving from variables in a quadratic equation to a time variable, and from changing and changing parameters to the coordinate, velocity and acceleration of the progression. In order for the content of the article to correspond to the topic, priority was given to mathematical initiatives (prerequisites, originals) of all the problems under consideration.

K-free and k-full numbers with prime factors in an arithmetic progression

We apply a classical Wirsing's theorem and study k-free numbers, k-full numbers and perfect powers with prime factors in an arithmetic progression.

Pavan Gampala's Pattern: A Novel Observation in Arithmetic Sequences

In this paper, we present a newly observed pattern in the sums of consecutive natural numbers. The pattern demonstrates that the sum of the first nnn natural numbers, when added to the square of nnn, equals the sum of the next nnn natural numbers. This finding introduces a unique relationship within arithmetic sequences, offering a fresh perspective on the properties of natural number summation. The implications of this pattern may extend to various areas of number theory, combinatorics, and mathematical analysis.

Optimization of relief hole blasting satisfying synergistic constraints of rock-breaking area and hole-bottom minimum burden

A strategy to optimize relief hole positions is proposed to enhance the effectiveness of tunnel blasting in rock breaking. This study employed two predefined arithmetic progressions to coordinate the distribution of rock-breaking areas and hole-bottom minimum burdens allocated to each row of holes. Iterative calculations were performed to determine the final position of each row of holes. To determine the optimal drilling scheme, modeling and simulation were conducted using a finite element method-smoothed particle hydrodynamics (FEM-SPH) coupling algorithm. This approach allowed for a comparison of the rock-breaking effects of relief holes under different constraint combinations. This study indicates that the displacement of rock particles varies in different depth zones. The largest displacements of the rock particles were observed in the middle of the charge section. For a conventional working face with a width of 12.2 m and a designed advance of 3 m, the rock-breaking efficiency is optimal when the two control ratios, rA and rW, are 1.5 and 1.4, respectively. This study advances the underground blasting design technology and contributes to energy reduction and efficiency improvements in blasting engineering.

The Prime Geodesic Theorem in Arithmetic Progressions

Abstract We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

Degree lowering for ergodic averages along arithmetic progressions

Degree lowering for ergodic averages along arithmetic progressions

On Orthogonal Polynomials Related to Arithmetic and Harmonic Sequences

On Orthogonal Polynomials Related to Arithmetic and Harmonic Sequences

Recurrent Relationships as an Important Class of Mathematical Equations in Chemistry

The unique potential and different areas of applications of recurrent (synonymous: recursive) relationships in chemistry and chromatography are considered. Recurrent relations can be used in two forms: as functions of integer arguments, y(x+ 1) = ay(x) + b, and as functions of equidistant argument values, A(x+ Δx) = aA(x)+ b, Δx= const. The first form applies to all physicochemical properties of homologs in organic chemistry, because the number of carbon (and other) atoms in a molecule can be integer only. The second one applies to chemical variables depending on temperature, pressure, concentrations, etc., when the chemists should provide equal “steps” of their variations. Recurrent relations combine the properties of arithmetic and geometric progressions, which accounts for their unique approximation abilities. This was illustrated by approximating the number of isomers of alkanes, the boiling points of homologs (nonlinear dependencies), the melting points of homologs (alternation effects), the temperature dependence of the solubility of inorganic salts in water, and by revealing the anomalies of gas chromatographic retention indices and retention times in reversed-phase high performance liquid chromatography.

Discrete Ω-results for the Riemann zeta function

Abstract We study lower bounds for the Riemann zeta function ζ ⁢ ( s ) {\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s ) {\zeta(s)} .