Non-Extendability of Special DIO 3-Tuples Involving Nonaganal Pyramidal Number

Abstract

This paper concerns with the construction of three distinct polynomials with integer coefficients (a1, a2, a3) such that the product of any two contribution of the set subtracted to their sum and improved by a non-zero integer (or a polynomial with integer coefficients) is a perfect square and this shows the non-extendability of Special Dio Quadruple.

Introduction

I. INTRODUCTION

Diophantine Analysis is the mathematical study of Diophantine Problems, which was initiated by Diophantus in third century. A set of m distinct positive integers {a1, a2, …, am} is said to have the property D(n) if the product any two members of the set is decreased by their sum and increased by a non-zero integer n, is a perfect square for all m elements. Such a set is called Diophantine m-tuples of size m. Many mathematicians considered the extension problem of Diophantine quadruples with the property

D(n) for any arbitrary integer n  and also for any linear polynomial.

In this communication, we have presented three sections, in each of which we find the Diophantine triples for nonagonal Pyramidal number with distinct ranks and the non-extendability of Special Dio quadruple.

Conclusion

In this paper, we have presented the construction of a special dio 3-tuples for Pyramidal number with suitable properties and the non-extendability of Special Dio Quadruple. To conclude that one may search for Special Dio 3- tuples for higher order Pyramidal number with their corresponding suitable properties.

References

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Copyright

Copyright © 2023 S Vidhya, T Gokila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.