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Journal of Discrete Mathematical Sciences and Cryptography cover

Journal of Discrete Mathematical Sciences and Cryptography

(Since 1998)
Hybrid ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

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Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Exploring integral solutions for ternary Diophantine equations : A multi-technique approach

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pp. 2279–2288Vol. 28Issue 6September 2025DOI: 10.47974/JDMSC-2224 Crossmark XML
Received:
10 Dec 2024
Published Online:
16 Sep 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2224
Pages:
2279–2288

Abstract

This research paper deals with the problem of finding non-zero integral solutions for a ternary Diophantine equation expressed as pw2 – (2p – 1)wz + pz2 = (k2 + 4p – 1)qr. The investigation explores three distinct cases, each examining the exponent r from 1 to 3. Additionally, the paper outlines methodologies for finding solutions when r is greater than 3. The paper tailors a theorem to this ternary equation, utilizing seven unique techniques to address the problem. A crucial aspect of these methodologies involves a transformative approach where w is replaced with x + y and z is replaced with x – y. Through rigorous analysis and the application of these techniques, the paper provides comprehensive insights into the solutions of the aforementioned Diophantine equation, supported by numerical examples and graphical representations.

Keywords

Cubic equationFigurate numbersHigher degree Diophantine equationIntegral solutions

Subject Classifications

Primary 11D0411D0911D25Secondary 11D4111D72

References

[1] R. D. Carmichael, The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1959).
[2] L. E. Dickson, History of Theory of Numbers,  Chelsea Publishing Company, Vol.11, New York (1952).
[3] Elena Deza and Michel Marie Deza, Figurate Numbers,  World Scientific Publishing Co. Pte.Ltd, Singapore (2012).
[4] L. J. Mordell, Diophantine equations,  Academic Press, London, 1969. S. G. Telang, Number theory,  Tata Mc Graw Hill publishing company, New Delhi (1996).
[5] A. Priya and S. Vidhyalakshmi, On the Non-Homogeneous Ternary Quadratic Equation 2(x2 + y2) – 3xy + (x + y) + 1 = z2,  Journal of Mathematics and Informatics, vol.10, pp.49-55 (2017).  
[6] C. Saranya and P. Kayathri, Integral Solutions of the Ternary Cubic Equation 6(x2 + y2) – 11xy = 288z3,  International Journal for Research in Applied Science and Engineering Technology (IJRASET), vol. 10, no.I, pp.62-65 (2022).
[7] Chikkavarappu gnanendra rav, On The Ternary Quadratic Diophantine Equation 2x2 – 3xy + 2y2 = 56z2,  International Research Journal of Engineering and Technology (IRJET), vol. 05, no. 02, pp. 329-332 (2018).
[8] D. Maheshwari, S. Devibala and M. A. Gopalan, Integral Solutions of Cubic Diophantine Equation With Five Unknowns Simple Form For Coefficients x3 + y3 = 13(z + w)p2,  Advances and Applications in Mathematical Sciences, vol. 21, no. 8, pp. 4595-4604 (2022).
[9] G.Janaki and C.Saranya, Observations on Ternary Quadratic Diophantine Equation 6(x2 + y2) – 11xy + 3x + 3y + 9 = 72z2,  International Journal of Innovative Research in Science, Engineering and Technology, vol. 5, no. 2, pp. 2060-2065 (2016).
[10] G.Janaki and C.Saranya, On the Ternary Quadratic Diophantine Equation 5(x2 + y2) – 6xy = 4z2,  Imperial Journal of Interdisciplinary Research (IJIR), vol. 2, no. 3, pp. 396-397 (2016).
[11] G.Janaki and C.Saranya, Integral Solutions of Ternary Cubic Equation 3(x2 + y2) – 4xy + 2(x + y + 1) = 972z3,  International Research Journal of Engineering and Technology, vol. 04, no. 3, pp. 665-669 (2017).
[12] G.Janaki and S. Shanmuga Priya, An Analysis on the Ternary Cubic Diophantine Equation 2(l2 + m2) – 3lm = 56t3,  International Journal for Research in Applied Science and Engineering Technology (IJRASET), vol.11, no. III, pp.1320-1322 (2023).
[13] K.Dhivya and T.R.Usha Rani, On the Non-Homogeneous Quadratic Equation with Five Unknowns x2 + xy – y2 – (z + w) = 10p2,  Journal of Mathematics and Informatics, vol. 10, pp. 41-48 (2017).
[14] K. Manikandan and R. Venkatraman, The Ternary Cubic Equation of the Integral Solutions 5(x2 + y2) – 6xy + 4(x + y + 1) = 6100z3, AIP Conference Proceedings, vol. 2852, pp.1-4 (2023).
[15] K. Manikandan and R. Venkatraman, Integral Solutions of The Ternary Diophantine Equation 945(x2 + y2) – 1889xy + 10(x + y) + 100 = pzq,  IAENG International Journal of Applied Mathematics, vol. 53, pp. 1657-1672 (2023).
[16] K.Meena,M.A. Gopalan,N.Thiruniraiselvi and D. Kanaka, On Ternary Quadratic Diophantine Equation 3(x2 + y2) – 4xy = 42z2,  International Journal of Research in Engineering and Applied Sciences, vol. 5, no. 6, pp. 174-179 (2015).
[17] K. Thangamalar, On Ternary Quadratic Diophantine Equation 9(x2 + y2) – 17xy + x + y + 1 = 39z2,  Indian Journal of Applied Research, vol.9, no.1, pp.35-36 (2019).
[18] M. A. Gopalan, S. Vidhyalakshmi and R. Maheswari, On Homogeneous Ternary Quadratic Diophantine Equation 4(x2 + y2) – 7xy = 16z2,  Scholars Journal of Engineering and Technology (SJET), vol. 2, no.5, pp. 676-680 (2014).
[19] M.A.Gopalan, S.Mallika and S.Vidhyalakshmi, Non-Homogeneous Cubic Equation with Three Unknowns,  International Journal of Engineering Sciences and Research Technology, vol. 3, no. 12, pp. 138-141 (2014).
[20] M. A. Gopalan and V. Geetha, Latice points on the homogeneous cone 7(x2 + y2) – 13xy = 28z2,  The International Journal of Science and Technology, vol. 2, no. 7, pp. 291-295 (2014).
[21] Manju Somanath, Radhika Das and V.A. Bindu, Lattice Points On The Homogeneous Cubic Equation With Four Unknowns x2–xy + y2 + 4w2 = 8z3,  International Journal of Research - Granthaalayah, vol.8, no.8, pp.135-139, 2020.
[22] N. Bharathi and S. Vidhyalakshmi, Observation on the Non-Homogeneous Ternary Quadratic Equation x2 – xy + y­ + 2 (x + y) + 4 = 12z2,  Journal of Mathematics and Informatics, vol. 10, pp. 135-140 (2017).
[23] P. Jayakumar and G. Shankarakalidoss, Lattice Points on the Homogeneous Cone 4(x2 + y2) – 3xy = 19z2,  International Journal of Science and Research (IJSR), vol. 4, no. 1, pp. 2053-2055 (2013).
[24] P.Jayakumar, V.Pandian and A.Nirmala, Lattice Points on the Homogeneous Cone 5(x2 + y2) – 9xy = 23z2, International Journal of Research in Engineering and technology, vol .5, no. 8, pp. 78-81 (2016).
[25] R. Anbuselvi and S. A. Shanmugavadivu, Integral Solutions of Ternary Cubic Equation 5(x2 + y2) – 9xy + x + y + 1 = 28z3,  International Educational Scientific Research Journal [IESRJ], vol. 2, no. 11, pp. 8-10 (2016).
[26] R. Anbuselvi and S. A. Shanmugavadivu, On the Non Homogeneous Ternary Quadratic Equation x2 + xy + y2 = 12z2, IOSR Journal of Mathematics (IOSR-JM), vol. 12, no. 1, pp. 75-77 (2016).
[27] R. Anbuselvi and K. Kannaki, On the Non Homogeneous Ternary Quadratic Equation 7(x2 + y2) – 13xy + x + y + 1 = 31z2, International Educational Scientific Research Journal [IESRJ], vol. 3, no. 2, pp. 64-69 (2017).
[28] Rahim Mahmoudvand, Hossein Hassani, Abbas Farzaneh and Gareth Howell, The Exact Number of Nonnegative Integer Solutions for a Linear Diophantine Inequality,  IAENG International Journal of Applied Mathematics, vol. 40, no. 1, pp. 1-5 (2010).
[29] R. Venkatraman and P. Jayakumar, On The Non Homogeneous Heptic Equation with Five Unknowns (x2 – y2)(5x2 + 5y2 – 8xy) = 13(X2 – Y2)z3, Global Journal of Pure and Applied Mathematics (GJPAM), vol. 12, no. 2, pp. 361-363 (2016).
[30] R. Venkatraman, P. Jayakumar, S. Thalapathiraj and J. Ravikumar, On Heptic Diophantine Equations with three unknowns 21(x2 + y2) – 41xy = 99z7, AIP Conference Proceedings, vol. 2282, no. 020033, pp. 1-3 (2020).
[31] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, Observations on x2 + y2 + 2(x + y) + 2 = 10z2, Malaya Journal of Matematik, vol. 6, no. 3, pp. 632-634 (2018).
[32] S.Thenmozhi and S.Vidhyalakshmi, On the Ternary Quadratic Diophantine Equation 3(x2 + y2) – 5xy = 75z2, Journal of Mathematics and Informatics, vol. 10, pp. 11-19 (2017).
[33] S. Vidhyalakshmi, T.R. UshaRani and M.A.Gopalan, Integral Solutions of Ternary Cubic Equation 5(x2 + y2) – 9xy + x + y + 1=35z3,  IJRET: International Journal of Research in Engineering and Technology, vol. 03, no. 11, pp. 449-452 (2014).
[34] S. Vidhyalakshmi, T.R.Usharani and K.Hemalatha, On the ternary quadratic diophantine equation (x2 + y2) – xy + x + y + 1 = 16z2, International Journal of Engineering Sciences and Management Research, vol. 2, no. 7, pp. 87-92 (2015).
[35] S. Vidhyalakshmi and M.A. Gopalan, On finding integer solutions to non homogeneous ternary cubic Equation x2 + xy + y2 = (m2 + 3n2)z3, Journal of Advanced Education and Sciences, vol. 2, no. 4, pp. 28-31 (2022).

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