Perpetuation of Eminent Forms of 2-Tuples into 3-Tuples Interlacing Some k-Polygonal Numbers with Appropriate Properties

International Journal of Mathematics Trends and Technology (IJMTT)
© 2022 by IJMTT Journal
Volume-68 Issue-8
Year of Publication : 2022
Authors : S. Saranya, V. Pandichelvi

How to Cite?

S. Saranya, V. Pandichelvi, "Perpetuation of Eminent Forms of 2-Tuples into 3-Tuples Interlacing Some k-Polygonal Numbers with Appropriate Properties," International Journal of Mathematics Trends and Technology, vol. 68, no. 8, pp. 167-176, 2022. Crossref,

In this manuscript, the patterns of 3-tuples(unmk, vnmk, wnmk), (vnmk,wnmk, xnmk), (wnmk, xnmk, ynmk), (xnmk, ynmk, znmk) etc, m=1,2,3,4,5 where the elements are some k - polygonal numbers such that the product of two numbers enlarged by certain numbers remains a perfect square are appraised. Also, the MATLAB program for the verification of all the patterns of 3-tuples sustaining dissimilar properties is exemplified.

Keywords : Diophantine triples, Integer sequence, Polygonal numbers.


[1] Ivan Niven, Hibert S, Zuckerman and Huge L, Montgomery, “An Introduction to the Theory of Numbers,” Fifth Ed., John Wiley & Sons Inc., 2004.
[2] R.D. Carmichael, “The Theory of Numbers and Diophantine Analysis,” Dover Publication, New York, 1959.
[3] M.N.Deshpande, “Families of Diophantine Triplets”, Bulletin of the Marathwada Mathematical Society, vol. 4, pp. 19-21, 2003.
[4] Y.Bugeaud, A.Dujella and M.Mignotte, “On the Family of Diophantine Triple {k-1, k+1, 16k3-4k}," Glasgow Math. J., vol. 49, pp. 333-344, 2007.
[5] M.A.Gopalan, K.Geetha and ManjuSomanath, “On Special Diophantine Triple”, Archimedes Journal of Mathematics, vol. 4, no.1, pp. 37-43, 2014.
[6] G.Janaki and S.Vidhya, “Construction of the Diophantine Triple Involving Stella Octangula Number”, Journal of Mathematics and Informations, vol. 10, pp. 89-93, 2017.
[7] G.Janaki and S.Vidhya, “Construction of the Diophantine Triple Involving Pronic number,” IJRASET, vol. 6, pp. 2201-2204, 2017.
[8] V.Pandichelvi and P.Sandhya, “The Patterns of Diophantine Triples Engross Cheldhiya Companion Sequences with Inspiring Properties”, Adalya Journal, vol. 9, no. 4, pp. 399-404, 2020.
[9] V.Pandichelvi and P.Sivakamasundari, “On the Extendibility of the Sequences of Diophantine Triples Into Quadruples Involving Pell Numbers,” International Journal of Current Advanced Research, vol. 6, no. 11, pp. 7197-7202, 2017.
[10] V.Pandichelvi and S.Saranya, “Classification of an Exquisite Diophantine 4-Tuples Bestow with an Order”, Malaya Journal of Mathematik, vol. 9, no. 1, pp. 612-615, 2021.
[11] M.A.Gopalan, and V. Geetha, "Sequences of Diophantine Triples," JP Journal of Mathematical Sciences, vol. 14, no. 1, pp. 27-39, 2015.
[12] M.A.Gopalan, and Sharadha Kumar, "On the Sequences of Diophantine 3-Tuples Generated through Euler and Bernoulli Polynomials," Tamap Journal of Mathematics and Statistics, pp. 1-5, 2019.
[13] M.A Gopalan, V. Geetha, and S. Vidhyalakshmi, "Dio 3-Tuples for Special Numbers-I," The Bulletin of Society for Mathematical Services and Standards, vol. 10, pp. 1-6, 2014.
[14] Filipin, Alan, Bo He, and Alain Togbé, "On a Family of Two-Parametric D (4)-Triples," Gazette Mathematically, vol. 47, no. 1, pp. 31- 51, 2012.
[15] Filipin, Alan, and Alain Togbé, "On the Family of Diophantine Triples," Mathematical Journal of the Pedagogical Academy of Nyíregyházi, New Series, vol. 25, no. 2, pp.145-153, 2009.
[16] Park, Jinseo, and June Bok Lee, "Some Family of Diophantine Pairs with Fibonacci Numbers," Indian Journal of Pure and Applied Mathematics, vol. 50, no. 2 , pp. 367-384, 2019.
[17] Zhang, Yifan, and G. Grossman, "On Diophantine Triples and Quadruples," Notes Number Theory Discrete Math, vol. 21, pp. 6-16, 2015.
[18] Bayad, Abdelmejid, et al., "On the Extensibility of the D (4)-Triple {k-2, k+2, 4k} Over Gaussian Integers," Notes Number Theory Discrete Math, vol. 23, pp 1-26, 2017.
[19] Meena K., et al., "Sequences of Special Dio-Triples," IJMTT, vol. 10, no. 1, pp. 43-46, 2014.
[20] Gopalan, M. A., K. Geetha, and ManjuSomanath, "Special Dio 3-Tuples," The Bulletin of Society for Mathematical Services and Standards, vol. 10, pp. 22-25, 2014.
[21] Radha, Ms R., and G. Janaki. "Special Dio 3-Tuples II for Hex Number," The International Journal of Analytical and Experimental Modal Analysis, vol. 11, no. 10, pp. 1308-1312, 2019.
[22] Srividhya, G., and T. Ragunathan, "Special D (k2+1) Dio-Quadruples Involving k-Jacobsthal and k-Jacobsthal Lucas Numbers," International Journal of Applied Engineering Research, vol. 13, no. 21, pp. 15362-15364, 2018.
[23] Cipu, Mihai, Alan Filipin, and Yasutsugu Fujita, "An Infinite Two-Parameter Family of Diophantine Triples," Bulletin of the Malaysian Mathematical Sciences Society, vol. 43, no. 1, pp. 481-498, 2020.
[24] Vijayasankar, A., Sharadha Kumar, and M. A. Gopalan, "Formulation of Curious Family of 3-Tuples," International Journal of Innovation Engineering and Science Research, vol. 4, no. 1, pp. 1-11, 2020.
[25] V.Pandichelvi and B. Umamaheswari, "Structure of Categories of Diophantine 3-Tuples Apropos (2j,3) Polygonal Numbers," vol. 21, no. 8, pp. 4681-4692 , 2022.
[26] Saranya C and B. Achya, "Special Dio 3-Tuples Involving Square Pyramidal Numbers," International Journal for Research in Applied Science & Engineering Technology, vol. 10, no. 3, pp. 891-895, 2022.