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1

An exponential diophantine equation on triangular numbers

Hamtat Abdelkader

Mathematica Applicanda

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2023

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Vol. 51, no. 1

99--107

EN

Triangular numbers have the property that the difference of two consecutive terms in a sequence of triangular numbers is equal to the index of the first one. They also have the property that the difference of the squares of consecutive numbers is equal to the third power of the index of the first number. The aim of the work is to investigate the analogous exponential Diophantine equation. We consider the equation: the difference of the x-th powers of a sequence of triangular numbers is equal to the y-power of the index of the first one, for some positive integers x,y. We show that the above equation has only solutions (x,y)=(1,1) and (2,3).

PL

Liczby trójkątne mają tę właściwość, że różnica dwóch kolejnych wyrazów ciągu liczb trójkątnych jest równa indeksowi pierwszego. Mają też tę właściwość, że różnica kwadratów kolejnych liczb jest równa trzeciej potędze indeksu pierwszej liczby. Celem pracy jest zbadanie analogicznego diofantycznego równania wykładniczego. Rozważmy równanie: różnica x-tych potęg ciągu liczb trójkątnych jest równa potędze y indeksu pierwszej, dla pewnych dodatnich liczb całkowitych x, y. Pokazujemy, że powyższe równanie ma tylko rozwiązania (x, y) = (1, 1) i (2, 3).

2

About special properties of the hidden structure of triangular numbers for immediate factorization

Samojluk Artur

Technical Sciences / University of Warmia and Mazury in Olsztyn

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2022

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nr 25(1)

35--57

EN

The factorization problem belongs to a group of problems important in the security of information systems and cryptography. The article describes a new number factorization algorithm designed based on numerical experiments. We present an extension of number factorization using triangular numbers features. The described algorithm can be used to increase the security of key generation for the RSA algorithm.

3

A compositional approach to two-stage Data Envelopment Analysis in intuitionistic fuzzy environment

Javaherian Nafiseh, Hamzehee Ali, Tooranloo Hossein Sayyadi

Operations Research and Decisions

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2021

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Vol. 31, No. 1

21--39

EN

Classical methods of data envelopment analysis operate by measuring the efficiency of decision- -making units (DMUs) compared to similar units, without taking their internal structure into account. However, some DMUs consist of two stages, with the first stage producing an intermediate product, which is then consumed in the second stage to produce the final output. The efficiency of this type of DMU is often measured using two-stage network data envelopment analysis. In real world, most data are vague. Therefore, the inputs and outputs of systems with vagueness data create uncertainty challenges for DMUs. As a result, when uncertainty appears, intuitionistic fuzzy sets can show more information than classical fuzzy sets. This paper presents a model of two-stage Network Data Envelopment Analysis based on intuitionistic fuzzy data, which measures the efficiency of the first and second stages of each DMU, and the overall efficiency measures based on the stage efficiencies.