Congruence properties of indices of triangular numbers multiple of other triangular numbers

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

Thue's equation as a tool to solve two different problems

A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.

Characterizations and Identities for Isosceles Triangular Numbers

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

An Improving the Occupational Risk Assessment in Industrial Enterprises

An occupational risk assessment is one of the main processes to assure a safe and healthy workplace. It was shown, that this process is particularly important in the cause of industrial enterprises, in which the number of accidents is the largest. In these enterprises, one of the most often practiced methods is the PN-N-18002 method. However, it was concluded this method has some limitations. They concern the way of assessing the risks in the traditional number scale, which is less precise than the fuzzy triangular number (using in FAHP). Therefore, the aim is to improve the process of assessment in industrial enterprises by integrated the PN-N-18002 method with the FAHP method (Fuzzy Analytic Hierarchy Process). The test of the proposed method was made for three machine operator positions used to aggregate extraction in one of the Podkarpacie enterprises. These positions were: loader operator (Ł-34), digger operator (CAT 323), dredge operator (300/250 KREBS 10/8). The concept of the method was to identify in a precise way what is the greatest extent danger to the operators of these workplaces. It was shown, that it is the work at height. It was concluded, that this method can be practice to risk assessment of other workplaces, among others from industrial enterprises. The originality is the integrated risk assessment method (PN-N-18002) with the fuzzy multicriteria decision method (FAHP) as part of achieving the precise results of risk assessment.

On Triangular Secure Domination Number

Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m). A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m) is called a secure domination number of graph T_m. A secure dominating number γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers.

Solution of A Cubic Equation with Triangular Number as Coefficients

While solving a cubic equation one root is always identified using trial and error method. Here in this paper first the interval in which the real root appears is found and the real root is identified using continued fraction method. It is illustrated by an equation having polygonal numbers as coefficients

Framework and Evolution of Task Orientation using Fuzzy and GA

The process of selecting employees in any organization is done by mostly old ways. There is a lack of hypothetical support in this method, and error is possible in the result. In the research paper presented, we have outlined the recruitment and selection process of employees with the help of Fuzzy Triangular Number. The selection process given here is completely different from the methods given earlier. To find the ideal employee here, we use the modified method of solving assignment problem. Using certain linguistic variables, we use robust ranking method here. The solution obtained from this method is minimized using the revised approach of assignment. This modified approach is simpler than the Hungarian method previously used by other authors. Using this modified solution, the minimum solution is achieved by reducing the number of recurring assignments. After assignment process, finally selection of employees is given using criterion of GA.