Indian Journal of Science and Technology

Abstract

Objectives: Diophantine m-tuples are sets of positive integers with the property that the product of any two elements in the set increased by n is a square. The objective of this paper is to analyse the extendibility of pairs of elements to dio — triples for the various choices of n. Method: The problem of the occurrence of Diophantine triples is examined by various Mathematicians. There is no universal method to find the extension of dio — triples with specific properties. Diophantine equations form a very major part of research in Number theory. One among the important equations is the Pell’s equation for which the general solutions can be derived. In this paper, the concept of Pell’s equations is implemented to construct Diophantine triples. Findings: In this paper, we have studied the extendibility of of elements in a commutative ring R with the property that added with quadratic and bi-quadratic polynomials, yields a perfect square. The elements are chosen as linear polynomials, polygonal numbers and centered polygonal numbers and are examined to form triples using different properties. Novelty: In this paper, we give some new examples of pairs of polynomial, polygonal and centered polygonal numbers and check for its extendibility as a triple. The examples illustrate various theoretical properties and construction for linear and quadratic polynomials. The Table 1, Table 2, Table 3 looks to be simple, but they require tremendous calculations and simplifications. MSC Classification Number: 11D09, 11D99 Keywords: Special dio ­ triples, Diophantine triples, Diophantine equations, Pellian Equation, Polygonal numbers, Centered polygonal numbers