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Ijraset Journal For Research in Applied Science and Engineering Technology

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Formation of Diophantine Triples Involving Heptagonal Pyramidal Numbers

Authors: C Saranya, P Chitra

DOI Link: https://doi.org/10.22214/ijraset.2023.49076

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Abstract

In this paper, we seek for three specific polynomials with integer coefficients to such an extent that the product of any two numbers expanded by a non-zero number (or polynomials with integer coefficients) is a perfect square.

Introduction

I. INTRODUCTION

A Diophantine equation is a polynomial equation in number theory where only the integer solutions are considered or searched for, typically with two or more unknowns [1-4]. The term "Diophantine" refers to the Greek mathematician Diophantus of Alexandria, who investigated similar situations and was one of the pioneers in introducing symbolism to variable-based mathematics in the third century.

The problem of the occurrence of Dio triples and quadruples with the property D(n) for any integer n as well as for any linear polynomial in n was studied by a number of mathematicians [5-8]. In this particular situation, one may turn to [9–16] for a thorough analysis of various difficulties on Diophantine triples. Half companion Diophantine triple sequences were studied in [17]. These results motivated us to examine for Diophantine triples with involving heptagonal pyramidal numbers. This paper aims at constructing Dio-Triples where the product of any two members of the triple with the addition of a non-zero integer or a polynomial with integer coefficients satisfies the required property. Also, we present three sections where in each of which we find the Diophantine triples from heptagonal pyramidal number of different ranks with their corresponding properties.

Conclusion

We have presented the Diophantine triples involving heptagonal pyramidal numbers. To conclude one may look for triples or quadruples for different numbers with their relating properties.

References

[1] Beardon, A. F. and Deshpande., M. N. “Diophantine triples”, The Mathematical Gazette, vol.86, pp. 258-260, 2002. [2] Bugeaud, Y. Dujella, A. and Mignotte, M., “On the family of Diophantine triples” {K – 1, K + 1, 16 4K}, Glasgow Math. J. Vol. 49, pp.333 - 344, 2007. [3] Carmichael, R. D., “Theory of numbers and Diophantine analysis”, Dover Publications. [4] Deshpande, M. N., “Families of Diophantine triples”, Bulletin of the Marathwada Mathematical Society, Vol. 4, pp. 19 - 21. 2003. [5] Fujita, Y., “The extendability of Diphantine pairs ?k ?1, k ?1? ”, Journal of Number. [6] Gopalan, M.A. and Srividhya, G., “Two Special Diophantine Triples”, Diophantus Journal of Mathematics, Vol, 1 (2012), (accepted). [7] Hua, L.K., “Introduction to the Theory of Numbers”, Springer-Verlag, Berlin-New york, 1982. [8] Janaki, G. and Vidhya, S., “Construction of the Diophantine triple involving stella octangular number, Journal of Mathematics and Informatics, vol.10, Special issue, 89-93, Dec 2017. [9] Vidhya S, Janaki G., “Special Dio 3-tuples for Pronic number-I”, International Journal for Research in Applied Science and Engineering Technology, 5(XI), 159-162, 2017. [10] Janaki G, Vidhya S., “Special Dio 3-tuples for Pronic number-II”, International Journal of Advanced Science and Research, 2(6), 8-12, 2017. [11] Janaki, G. and Saranya, C., “Construction of The Diophantine Triple involving Pentatope Number”, International Journal for Research in Applied Science & Engineering Technology, vol.6, Issue III, March 2018. [12] Janaki, G. and Saranya, C., “Special Dio 3-tuples for pentatope number”, Journal of Mathematics and Informatics, vol.11, Special issue, 119-123, Dec 2017. [13] Janaki, G. and Saranya, C., “Some Non-Extendable Diophantine Triples Involving Centered Square Number”, International Journal of scientific research in mathematical and statistical , Volume 6, Issue 6, Pg. No. 109-107, December 2019. [14] Saranya, C and Achya, B., “Special Diophantine triples involving square pyramidal number”, Indian Journal of advanced mathematics, volume 1,issue 2,October 2021. [15] Saranya, C and Achya, B., “Diophantine triples Involving Square Pyramidal Numbers”, Advances and Applications in mathematical sciences , Volume 21,Issue 3, pg. No.1541-1547, January 2022. [16] Saranya, C and Manjula, K., “Construction of Diophantine triples involving hexagonal Saranya, C and Janaki, G., pyramidal numbers”, International journal science research in mathematics and statistical sciences, volume 9, issue 4, 2022. [17] Saranya, C and Achya, B., “Special Dio 3-Tuples Involving Square Pyramidal Numbers”, International Journal for Research in Applied Science & Engineering Technology, vol.10, Issue III, March 2022. [18] Saranya, C and Janaki, G., “Half companion sequences of special dio 3-tuples involving centered square number”, International journal for recent technology and engineering, volume 8, issue 3, pages 3843-3845, September 2019.

Copyright

Copyright © 2023 C Saranya, P Chitra. 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 : IJRASET49076

Publish Date : 2023-02-11

ISSN : 2321-9653

Publisher Name : IJRASET

DOI Link : Click Here