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From the viewpoint of atomic bond relaxation, an analytical approach was put forward to elucidate the physical origins of crystal size and cross-sectional shape dependency of piezoelectric potentials in GaN nanowires and nanotubes. It is demonstrated that (i) size-induced increase in piezoelectric potential is attributed to the coupling effect of the rising piezoelectric coefficient and both the reducing dielectric constant and elastic constant caused by the surface atomic coordination number loss, bond energy perturbation, and surface-to-volume ratio rising; (ii) as the number of sides for polygonal nanowires or nanotubes with the same equivalent radius decreases, the surface-to-volume ratio rises, and the piezoelectric potential increases; and (iii) the nanotubes can generate a piezoelectric potential higher than their nanowire counterparts due to their larger surface-to-volume ratios. The proposed formulation offers a scientific basis for the fabrication, optimization, and modulation of one-dimensional GaN-based piezoelectric nanometer devices. [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of integers as sums of at most four distinct polygonal numbers with a prescribed number of repeats of each distinct polygonal number. We compare such representations with classical polygonal numbers, and those representations with generalized polygonal numbers. Our main result is that representations with classical polygonal numbers are equidistributed in the sense that the number of representations in the nonnegative quadrant in four-dimensional space is asymptotically 1 16 of the representations in the entire space. [ABSTRACT FROM AUTHOR]

Let P 8 (x) = 3 x 2 - 2 x . For positive integers a 1 , a 2 , ⋯ , a k , a polynomial of the form a 1 P 8 (x 1) + a 2 P 8 (x 2) + ⋯ + a k P 8 (x k) is called an octagonal form. For a positive integer n, an octagonal form is called tight T (n) -universal if it represents (over Z ) every positive integer greater than or equal to n and does not represent any positive integer less than n. In this article, we find all tight T (n) -universal octagonal forms for every n ≥ 2 . [ABSTRACT FROM AUTHOR]

We determine extremal graphs with respect to the number of independent sets for polygonal chains joined by cutting edges. We show that for any class of polygonal chains, independently of the number of sides on each polygon, the zig-zag polygonal chain has the extremal minimum value. Meanwhile, the polygonal chain at distance 2 between any two consecutive cutting edges provides the extremal maximum value. These results generalize and formalize previous ideas about phenylene chains. To recognize those extremal topologies, extremal values for the product between two Fibonacci numbers with complementary indices are determined. The product between two Fibonacci numbers with complementary indices are also applied for recognizing extremal graphs when a path is extended with one more vertex, or when any independent subgraph is linked to the path. Our results are achieved by applying decomposition of the input graph through the vertex and edge division rules. [ABSTRACT FROM AUTHOR]

In this paper, we investigate arithmetic progressions in integer-valued quadratic polynomials. We prove two results for integer-valued quadratic polynomials. First, we prove that there is no three-term arithmetic progression with a perfect square as a common difference for a class of integer-valued quadratic polynomials. We then find a class of numbers depending on integer-valued quadratic polynomials, which cannot work as a common difference for three-term arithmetic progression. Further, we deduce some results for arithmetic progressions in the sequence of polygonal and centered polygonal numbers. [ABSTRACT FROM AUTHOR]

In real world applications, data are subject to ambiguity due to several factors; fuzzy sets and fuzzy numbers propose a great tool to model such ambiguity. In case of hesitation, the complement of a membership value in fuzzy numbers can be different from the non-membership value, in which case we can model using intuitionistic fuzzy numbers as they provide flexibility by defining both a membership and a non-membership functions. In this article, we consider the intuitionistic fuzzy linear programming problem with intuitionistic polygonal fuzzy numbers, which is a generalization of the previous polygonal fuzzy numbers found in the literature. We present a modification of the simplex method that can be used to solve any general intuitionistic fuzzy linear programming problem after approximating the problem by an intuitionistic polygonal fuzzy number with n edges. This method is given in a simple tableau formulation, and then applied on numerical examples for clarity. [ABSTRACT FROM AUTHOR]

Let, be the sequence of polygonal numbers where, is an integer. In this paper, we describe two wave networks in cellular automata that can generate the sequences, in real-time. The first is a wave network that yields a cellular automaton with 5/2 + 2 states that generates this sequence when s = 4 is an even integer. With the second wave network, we obtain a cellular automaton with s states when s = 5 is an odd integer, or a cellular automaton with four states when s = 3. [ABSTRACT FROM AUTHOR]

The polygonal interval-valued fuzzy number is constructed based on the polygonal fuzzy number and the interval-valued fuzzy number. Its main feature is that the linear operation of finite ordered points reduces the complexity of traditional interval-valued fuzzy number operations. This research presents a generalized distance formula between two polygonal interval-valued fuzzy numbers and explores topological properties under the distance of polygonal interval-valued fuzzy numbers. In addition, we adopt the TOPSIS (technique for order preference by similarity to an ideal solution) and prospect theory approach for the multi-attribute decision-making problem. The information of attributes describes with polygonal interval-valued fuzzy numbers, and we then implement optimized ranking on the alternatives according to the profit and loss ratio. Finally, we verify the effectiveness and practicability of the decision-making method and fuzzy numbers at polygonal interval-valued fuzzy numbers in e-commerce risk assessment. [ABSTRACT FROM AUTHOR]

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy's lemma. Using this, we show the "pentagonal theorem of 63", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 11 , 13 , 14 , 17 , 18 , 19 , 23 , 28 , 31 , 33 , 34 , 39 , 42 , 63. We further show that these integers form a unique minimal universality criterion set. [ABSTRACT FROM AUTHOR]

Summary: We present visual proofs illustrating certain identities relating hexagonal numbers to other figurate numbers. [ABSTRACT FROM AUTHOR]

Copyright of Chinese Journal of High Pressure Physics is the property of Chinese Journal of High Pressure Physics Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The aim of this study was to investigate the involvement of the mesencephalic superior colliculus (SC) in the pathogenetic mechanism of SIDS, a syndrome frequently ascribed to arousal failure from sleep. We analyzed the brains of 44 infants who died suddenly within the first 7 months of life, among which were 26 infants with SIDS and 18 controls. In-depth neuropathological investigations of serial sections of the midbrain showed the SC layered cytoarchitectural organization already well known in animals, as made up of seven distinct layers, but so far never highlighted in humans, albeit with some differences. In 69% of SIDS cases but never in the controls, we observed alterations of the laminar arrangement of the SC deep layers (precisely, an increased number of polygonal cells invading the superficial layers and an increased presence of intensely stained myelinated fibers). Since it has been demonstrated in experimental studies that the deep layers of the SC exert motor control including that of the head, their developmental disorder could lead to the failure of newborns who are in a prone position to resume regular breathing by moving their heads in the sleep-arousal phase. The SC anomalies highlighted here represent a new step in understanding the pathogenetic process that leads to SIDS. [ABSTRACT FROM AUTHOR]

Integrating multiple independent functions into a single optical component is one of the most important topics in research on photoelectric systems. In this paper, we propose a multifunctional all-dielectric metasurface that can achieve a variety of non-diffractive beams depending on the polarization state of the incident light. Using the anisotropic TiO2 rectangular column as the unit structure, the three functions of generating polygonal Bessel vortex beams under left-handed circularly polarized incidence, Airy vortex beams under right-handed circularly polarized incidence and polygonal Airy vortex-like beams under linearly polarized incidence are realized. In addition, the number of polygonal beam sides and the position of focal plane can be adjusted. The device could facilitate further developments in scaling complex integrated optical systems and fabricating efficient multifunctional components. [ABSTRACT FROM AUTHOR]

Figurate numbers are numbers that can be represented by a regular and discrete geometric pattern of evenly-spaced points. Their study has attracted the attention of many mathematicians and scientists since the dawn of mathematical history, including Pythagoras of Samos (582 BC-507 BC), Diophantus of Alexandria (200/214-284/298), Fibonacci (1170-1250), Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), Waclaw Franciszek Sierpińshi (1882-1969) [1]. Although it is not one of the basic topics, it serves many fields of mathematics, such as Number Theory and Geometry. Many special numbers are related to figurate numbers. Polynomial values, some theorems and solutions of Diophantine equations are expressed in figurate numbers and studied [2, 3, 4]. Two-dimensional figurate numbers are known as polygonal numbers. In the early 1990's, polygonal numbers were expressed and visualised with the help of computers [5]. Richard K. Guy asked the question 'Every number is expressible as the sum of how many polygonal numbers?' [6]. As can be understood from his study, it is seen that most of the studies conducted are about ordinary polygonal numbers. But Euler proved the 'generalized pentagonal number theorem' about partitions [7]. [ABSTRACT FROM AUTHOR]

Minimal neighbour designs (NDs) are used when a response of a treatment (direct effect) is affected by the treatment(s) applied in the neighbouring units. Minimal generalised NDs are preferred when minimal NDs cannot be constructed. Through the method of cyclic shifts (Rule I), the conditions for the existence of minimal circular generalised NDs are discussed, in which v/2 unordered pairs do not appear as neighbours. Certain generators are also developed to obtain minimal circular generalised NDs in blocks of two different sizes, where k2 = 3, 4 and 5. All these designs are constructed using i sets of shifts for k1 and two for k2. [ABSTRACT FROM AUTHOR]

Figure 3 illustrates this equation for HT ht I . i The case k I = i 7, HT ht , for HT ht . Notation The I k i -gonal number with I n i sides HT ht can be defined with reference to triangular numbers HT ht via the equation HT ht for HT ht [[2]]. [Extracted from the article]

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For a positive integer n , let $\mathcal T(n)$ denote the set of all integers greater than or equal to n. A sum of generalised m -gonal numbers g is called tight $\mathcal T(n)$ -universal if the set of all nonzero integers represented by g is equal to $\mathcal T(n)$. We prove the existence of a minimal tight $\mathcal T(n)$ -universality criterion set for a sum of generalised m -gonal numbers for any pair $(m,n)$. To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$ -universal sums of generalised m -gonal numbers. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$ -universal sums of generalised m -gonal numbers. [ABSTRACT FROM AUTHOR]

'Slide scanners' are rapid optical microscopes equipped with automated and accurate x-y travel stages with virtual z-motion that cannot be rotated. In biomedical microscopic imaging, they are widely deployed to generate whole-slide images (WSI) of tissue samples in various modes of illumination. The availability of WSI has motivated the development of instrument-agnostic advanced image analysis software, helping drug development, pathology, and many other areas of research. Slide scanners are now being modified to enable polarised petrographic microscopy by simulating stage rotation with the acquisition of multiple rotation angles of the polariser–analyser pair for observing randomly oriented anisotropic materials. Here we report on the calibration strategy of one repurposed slide scanner and describe a pilot image analysis pipeline designed to introduce the wider audience to the complexity of performing computer-assisted feature recognition on mineral groups. The repurposed biological scanner produces transmitted light plane- and cross-polarised (TL-PPL and XPL) and unpolarised reflected light (RL) WSI from polished thin sections or slim epoxy mounts at various magnifications, yielding pixel dimensions from ca. 2.7 × 2.7 to 0.14 × 0.14 µm. A data tree of 14 WSI is regularly obtained, containing two RL and six of each PPL and XPL WSI (at 18° rotation increments). This pyramidal image stack is stitched and built into a local server database simultaneously with acquisition. The pyramids (multi-resolution 'cubes') can be viewed with freeware locally deployed for teaching petrography and collaborative research. The main progress reported here concerns image analysis with a pilot open-source software pipeline enabling semantic segmentation on petrographic imagery. For this purpose, all WSI are post-processed and aligned to a 'fixed' reflective surface (RL), and the PPL and XPL stacks are then summarised in one image, each with ray tracing that describes visible light reflection, absorption, and O- and E-wave interference phenomena. The maximum red-green-blue values were found to best overcome the limitation of refractive index anisotropy for segmentation based on pixel-neighbouring feature maps. This strongly reduces the variation in dichroism in PPL and interference colour in XPL. The synthetic ray trace WSI is then combined with one RL to estimate modal mineralogy with multi-scale algorithms originally designed for object-based cell segmentation in pathological tissues. This requires generating a small number of polygonal expert annotations that inform a training dataset, enabling on-the-fly machine learning classification into mineral classes. The accuracy of the approach was tested by comparison with modal mineralogy obtained by energy-dispersive spectroscopy scanning electron microscopy (SEM-EDX) for a suite of rocks of simple mineralogy (granulites and peridotite). The strengths and limitations of the pixel-based classification approach are described, and phenomena from sample preparation imperfections to semantic segmentation artefacts around fine-grained minerals and/or of indiscriminate optical properties are discussed. Finally, we provide an outlook on image analysis strategies that will improve the status quo by using the first-pass mineralogy identification from optical WSI to generate a location grid to obtain targeted chemical data (e.g., by SEM-EDX) and by considering the rock texture. [ABSTRACT FROM AUTHOR]

Recently, Jha has found identities that connect certain sums over the divisors of n to the number of representations of n as a sum of squares and triangular numbers. In this note, we state a generalized result that gives such relations for s-gonal numbers for any integer s ≥ 3. [ABSTRACT FROM AUTHOR]

We determine all natural numbers that can be expressed as the difference of two m-gonal numbers. For each such number, we determine the number of possible expression as the difference of two m-gonal numbers. [ABSTRACT FROM AUTHOR]

Given a set S of n points, a weight function w to associate a non-negative weight to each point in S , a positive integer k ≥ 1 , and a real number > 0 , we present algorithms for computing a spanner network G (S , E) for the metric space (S , d w) induced by the weighted points in S. The weighted distance function d w on the set S of points is defined as follows: for any p , q ∈ S , d w (p , q) is equal to w (p) + d π (p , q) + w (q) if p ≠ q , otherwise, d w (p , q) is 0. Here, d π (p , q) is the Euclidean distance between p and q if points in S are in ℝ d , otherwise, it is the geodesic (Euclidean) distance between p and q. The following are our results: (1) When the weighted points in S are located in ℝ d , we compute a k -vertex fault-tolerant (4 +) -spanner network of size O (k n 2 ). (2) When the weighted points in S are located in the relative interior of the free space of a polygonal domain , we detail an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network with O (k n h + 1 2 lg n) edges. Here, h is the number of simple polygonal holes in . (3) When the weighted points in S are located on a polyhedral terrain , we propose an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network, and the number of edges in this network is O (k n 2 lg n). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, basic information about centered polygonal numbers and continuous fractions are given. By using the continuous fraction algorithm, the relations between centered polygonal numbers and continued fractions have been studied. A general form for continued fractions of ratios of centered polygonal numbers of consecutive order is explained. [ABSTRACT FROM AUTHOR]

The application of wire EDM to the machining of cylindrical parts is an emerging research topic. The current research is mainly focused on the processing of microcylindrical parts, and the processing removal of microparts is usually small. In the complete machining process, especially during rough machining, the removal amount of the workpiece is usually large. In recent years, there have been few studies in this area. This paper proposes the application of WEDM with contour approximation method (CAM-WEDM) to the machining of cylindrical parts with large removal volume. First, the basic principle of the contour approximation method is introduced, and the machining process is divided into three parts: roughing, semi-finishing and finishing. Roughing corresponds to polygon cutting. Second, the various indicators of polygon cutting are analyzed. The influence of the number of polygonal sides on the residual height and residual area is analyzed. Two methods are proposed for polygon cutting: sequential cutting and multiple cutting. The calculation formulas for the machining volume of the two methods are deduced. By comparing the two methods, it is found that different methods are applicable to different part sizes. At last, machining experiments were carried out. The selection of residual variables and the determination of polygon machining methods are detailed. The process parameters and machining time of each stage are listed. In addition, the measurement results of each stage of the machining are also analyzed. At the same time, by comparing with the general WEDT, it is concluded that CAM-WEDM can effectively process cylindrical parts with large removal volume. [ABSTRACT FROM AUTHOR]

As a special case of general fuzzy numbers, the polygonal fuzzy number can describe a fuzzy object by means of an ordered representation of finite real numbers. Different from general fuzzy numbers, the polygonal fuzzy numbers overcome the shortcoming of complex operations based on Zadeh's traditional expansion principle, and can maintain the closeness of arithmetic operation. Hence, it is feasible to use a polygonal fuzzy number to approximate a general fuzzy number. First, an extension theorem of continuous functions on a real compact set is given according to open set construction theorem. Then using Weierstrass approximation theorem and ordered representation of the polygonal fuzzy numbers, the convergence of a single hidden layer feedforward polygonal fuzzy neural network is proved. Secondly, the gradient vector of the approximation error function and the optimization parameter vector of the network are given by using the ordered representation of polygonal fuzzy numbers, and then the gradient descent algorithm is used to train the optimal parameters of the polygonal fuzzy neural network iteratively. Finally, two simulation examples are given to verify the approximation ability of the network. Simulation result shows that the proposed network and the gradient descent algorithm are effective, and the single hidden layer feedforward network have good abilities in learning and generalization. [ABSTRACT FROM AUTHOR]

In preschool, numbers and shapes typically appear as separate topics. This study explores how a game, designed as a guided play activity with figurate numbers, functions in a preschool context. The guided play involved parking Lego cars in a rectangular shape, and to find out for which number of Lego cars this is possible. Thirteen preschool children in three separate age groups, aged from four to six years, together with their teacher, participated in the study. Their communications through words and actions were recorded. The results exemplify how this guided play provides a rich context for engaging young children with mathematical activities such as counting, sorting, shaping, asking, justifying, and inferring, as well as emotional engagement with the activity. [ABSTRACT FROM AUTHOR]

For a subset S of nonnegative integers and a vector $\mathbf {a}=(a_1,\ldots ,a_k)$ of positive integers, define the set $V^{\prime }_S(\mathbf {a})=\{ a_1s_1+\cdots +a_ks_k : s_i\in S\}-\{0\}$. For a positive integer n, let $\mathcal T(n)$ be the set of integers greater than or equal to n. We consider the problem of finding all vectors $\mathbf {a}$ satisfying $V^{\prime }_S(\mathbf {a})=\mathcal T(n)$ when S is the set of (generalised) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers. [ABSTRACT FROM AUTHOR]

Some convolution-type identities involving polygonal numbers and Horadam numbers are derived. The method of proof is to properly relate the generating functions to each other. Additionally, we prove a general non-convolutional result involving these number families and discuss some of the consequences.

Merca ['Congruence identities involving sums of odd divisors function', Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca's work, we complete the proof of these conjectures. [ABSTRACT FROM AUTHOR]

Let $\mathcal {P}_s(n)$ denote the nth s-gonal number. We consider the equation $$ \begin{align*}\mathcal{P}_s(n) = y^m \end{align*} $$ for integers $n,s,y$ and m. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$ , that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number greater than 1 expressible as a perfect mth power with $m>1$ is $\mathcal {P}_{10}(3) = 3^3$. [ABSTRACT FROM AUTHOR]

Through this article, we will provide a new proof of the below theorem Theorem: The equation P(s, x) + P(s, y) = P(s, z) has infinitely many solutions in positive integers (x, y, z) for any given positive integer s ≥ 3, where P(r, k) is the kthr-gonal number. Also, we will provide the syntax of a computer program written in Python, which will help us find the infinitely many solutions of the above theorem according to the choice of some variables, which we will introduce in this paper. [ABSTRACT FROM AUTHOR]

The concept of Figurate numbers is very ancient but it continues to fascinate amateur and professional mathematicians to this day. The connection between the two oldest branches of mathematics namely number theory and geometry was made possible in the study of Figurate numbers. In this paper, after introducing Figurate numbers formally, we had defined a new concept which is identified as Cesaro - Like summation and apply it for the family of Figurate numbers in two kinds. These ideas will pave a new way for deep understanding and further generalizations in summability theory. [ABSTRACT FROM AUTHOR]

It is an integer at least for HT ht up to 42. Two players take turns to press keys which must be different from, but in the same row or column as, the previous player; the first player whose keys sum to more than a given integer HT ht losing. K$ ht is the product of the first HT ht primes. The 15 cases of HT ht with integer sidelengths at most 30,000 are listed. Richard's last paper [129] co-authored with some computer scientists, provided much stronger numerical evidence of this, including investigating 8000 sequences with randomly selected initial numbers until they terminated or exceeded HT ht . [Extracted from the article]

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs. [ABSTRACT FROM AUTHOR]

Balancing numbers are certain integers defined relative to a Diophantine equation between triangular numbers. Many properties, relations and generalizations of balancing numbers have appeared in the literature. We present, in two parts, the basic theory of polygonal balancing numbers. In this part, we define s-agonal-balancing, s-agonal Lucas-balancing and s-agonal-counterbalancing numbers in terms of three conditions. The first condition requires s-agonal-balancing numbers and s-agonal Lucas-balancing numbers to be among the solutions of a Diophantine Pell equation for s ≥ 3. Groups of transformations that act on solutions of the s-agonal-balancing Diophantine equation are defined, and the orbits of solutions under the group actions are determined. Moreover, connections among the solutions of the family of s-agonal-balancing numbers for all values of s will be demonstrated. These connections allow us to delineate all solutions of the s-agonal-balancing Diophantine equations. [ABSTRACT FROM AUTHOR]

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number m has the unique property that sp(m) = p holds for each prime factor p, where sp(m) is the sum of the base-p digits of m. The first such number is Ramanujan's famous taxicab number 1729. Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials U3(t) ∈ Z[t], the simplest one being U3(t) = (6t + 1)(12t + 1)(18t + 1). We show that the values of any U3(t) obey a special decomposition for all t ≥ 2 and besides certain exceptions also in the case t = 1. These cases further imply that if all three factors of U3(t) are simultaneously odd primes, then U3(t) is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that sp(m) = p holds for the greatest prime factor p of m. Subsequently, we show some connections to taxicab and polygonal numbers, involving the number 1729 as an example again. [ABSTRACT FROM AUTHOR]

In this paper, some information about polygonal numbers are given. Also, a general binary operator that includes all polygonal numbers are given and it is investigated whether the algebraic structures defined with the general operator specify a semigroup or not. [ABSTRACT FROM AUTHOR]

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as x 2 + y 2 + z 2 for some integers x , y , z ≠ ± 1. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when k = 3 , which are sums of k nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer k ≥ 3. [ABSTRACT FROM AUTHOR]

Summary–Historical problems related to cycloids form the background for an investigation of paths traced by rolling regular polygons. Using trigonometry, geometry, and calculus, the lengths of and areas under the generated paths are shown to converge to values associated with the classical cycloid as the number of polygonal sides increases. [ABSTRACT FROM AUTHOR]