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ISSN: 2321-9653
Estd : 2013
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Ijraset Journal For Research in Applied Science and Engineering Technology

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

Authors: S Vidhya, T Gokila

DOI Link: https://doi.org/10.22214/ijraset.2023.49096

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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

[1] Carmichael R.D, History of Theory of numbers and Diophantine Analysis, Dover Publication, New York, 1959. [2] Mordell L.J, Diophantine equations, Academic press, London, 1969. [3] Nagell T, Introduction to Number Theory, Chelsea publishing company, New York, 1982. [4] Brown E, “Sets in which xy ? k is always a perfect square”, Math. Comp, 1985, 45, 613-620 [5] Beardon A.F, Deshpande M.N, “Diophantine Triples”, the Mathematical Gazette, 2002, 86, 258-260. [6] Vidhya S, Janaki G (2017), Special Dio 3-tuples for Pronic number-I, International Journal for Research in Applied Science and Engineering Technology, 5(XI), 159-162. [7] Janaki G, Vidhya S (2017), Special Dio 3-tuples for Pronic number-II, International Journal of Advanced Science and Research, 2(6), 8-12. [8] Vidhya S, Janaki G (2019), Elevation of Stella Octangula number as a Special Dio 3-tuples and the non-extendability of special Dio quadruple, Adalya Journal, 8(8), 621-624. [9] Janaki G, Saranya C (2017), Special Dio 3-tuples for Pentatope number, Journal of Mathematics and Informatics, 11, 119-123. [10] Gopalan M.A, Geetha V and Vidhyalakshmi S (2014), Special Dio 3-tuples for Special Numbers-I, The Bulletin of Society for Mathematical Services and Standards, 10, 1-6. [11] Gopalan M.A, Geetha K and Somnath M (2014), Special Dio 3-tuples, The Bulletin of Society for Mathematical Services and Standards, 10, 22-25.

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.

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Paper Id : IJRASET49096

Publish Date : 2023-02-13

ISSN : 2321-9653

Publisher Name : IJRASET

DOI Link : Click Here