Patterns of Integer Solutions to Non-Homogeneous Ternary Nonic Equation
Abstract
This paper aims at determining patterns of non-zero distinct integer solutions to nonhomogeneous
ternary nonic Diophantine equation x^2 + y^2 = 125z^9. Substitution technique and factorization method are
utilized to obtain the same. We are able to solve the given higher degree equation through the methods
suggested above for obtaining plenty of non-zero distinct integer solutions.
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References
M.A.Gopalan ,S.Vidhyalakshmi ,J.Shanthi ,On the Cubic Equation with Four Unknowns ,International Journal of Mathematics Trends andTechnology,Vol 20(1),Pg 75-84,2015.
M.A.Gopalan,S.Vidhyalakshmi,J.Shanthi, On ternary cubic Diophantine equation
3(𝑥2+ 𝑦2) − 5𝑥𝑦 + 𝑥 + 𝑦 + 1 = 12𝑧3, International Journal of Applied Research, vol1(8), 209-212,2015.
M.A.Gopalan ,S.Vidhyalakshmi,J.Shanthi ,On Cubic Equation with Four Unknowns ,International Journal for Mathematics ,Vol 2(3),
Pg 1-8,2016.
M.A.Gopalan,S.Vidhyalakshmi,J.Shanthi ,On The Non-homogeneous Cubic Equation with Five Unknowns ,International Journal of Information Research and Review (IJIRR), Vol 3(6),Pg 2525-2528,2016.
M.A.Gopalan,J.Shanthi,On The Non-homogeneous Cubic Equation with Five Unknowns ,International Journal of Modern Sciences and Engineering Technology (IJMSET), Vol 3(5),Pg 32-36,2016.
E.Premalatha, J.Shanthi, M.A.Gopalan, On Non - Homogeneous Cubic Equation with Four Unknowns , Vol.14, Issue 5, 126-129, March 2021.
J.Shanthi,M.A.Gopalan, A search on Non –distinct Integer solutions to cubic Diophantine equation with four unknowns x^2-xy+y^2+4w^2=8z^3, International Research Journal of Education and Technology,(IRJEdT), Volume2,Issue01, 27- 32, May 2021.
S.Vidhyalakshmi, J.Shanthi, M.A.Gopalan, “On Homogeneous Cubic equation with four
Unknowns , International Journalof Engineering
Technology Research and Management, 5(7), 180-185, July 2021.
S.Vidhyalakshmi,J.Shanthi, M.A.Gopalan, T. Mahalakshmi, “On the non-homogeneous Ternary Cubic Diophantine equationw^2-z^2+2wx-2zx=x^3”, International Journal of Engineering Applied Science &Technology, Vol-7, Issue-3, 120-121, July-2022,.
M.A. Gopalan, J. Shanthi, V.Anbuvalli, Obervation on The Paper Entitled Solutions of the Homogeneous Cubic Eqution with Six Unknowns ,International Journal of Research Publication&Reviews, Vol-4, Issue-2, 313-317, Feb-2023.
J.Shanthi, S.Vidhyalakshmi,M.A.Gopalan, On Homogeneous Cubic Equation with Four
Unknowns(x^3+y^3 )=7zw^2,” Jananabha, Vol-53(1), 165-172, May-2023.
J. Shanthi , M.A. Gopalan, Cubic Diophantine equation of the form
Nxyz=w(xy+yz+zx) , International Journal of Modernization in Engineering Tech &Science, Vol-5, Issue-9, 1462-1463, Sep-2023.
J. Shanthi, M.A. Gopalan,”A Search on Integral Solutions to the Non- Homogeneous
Ternary Cubic Equationax^2+by^2=(a+b)z^3,a,b>0”, International Journal of Advanced Research in Science, Communication and Technology, Vol-4, Issue-1, 88-92, Nov-2024.
J.Shanthi,M.A.Gopalan,On finding Integer Solutions to Binary Cubic Equation ,International Journal of Multidisciplinary Research in Science, Engineering and Technology ,7(11), 16816-16820,2024.
J.Shanthi,M.A.Gopalan ,A Classification of Integer Solutions to Binary Cubic Equation ,International Journal of Progressive Research in Engineering Management and Science (IJPREMS) ,5(5),825-1828, 2025.
J.Shanthi,M.A.Gopalan,Observations on Binary Cubic Equation , IJARSCT, 5(1), 1-5, 2025.
J.Shanthi, M.A.Gopalan,On Solving Binary Cubic Equation , IRJEdT, 8(6), 139-144, 2025.
J.Shanthi, M.A.Gopalan, On finding Integer Solutions to Binary Quintic Equation , International Research Journal of Education andTechnology, 6(10), 226-231, 2024.
Sharadha Kumar, M.A.Gopalan, “On the Cubic Equation ”, JETIR, 6(1), 658-660, January 2019.
M.A.Gopalan, Sharadha Kumar, “On the non-homogeneous ternary cubic equation ”, International Journal of Engineering and Techniques, 4(5), 105-107, Sep-Oct (2018).
J.Shanthi, M.A.Gopalan, On finding Integer Solutions to Binary Quintic Equation , International Journal of Advanced Research in Science,Engineering and Technology, 11(10), 22369-22372, 2024.
J.Shanthi, M.A.Gopalan, Delineation of Integer Solutions to Non-Homogeneous QuinaryQuintic Diophantine Equation , International Journal of Research Publication and Reviews, 4(9), 1454-1457, 2023.
J.Shanthi,M.A.Gopalan, On the Ternary Non-homogeneous Quintic Equation , International Journal of Research Publication and Reviews, 4(8), 329-332, 2023.
J.Shanthi, M.A.Gopalan, On the Non-homogeneous Quinary Quintic Equation , International Research Journal of Education and Technology, 05 (08),238-245, 2023.
N.Thiruniraiselvi, M.A.Gopalan, The Non-Homogeneous Quintic Equation with Six Unknowns , International Journal for Research in Applied Science & Engineering Technology ,7(VI), 2527-2530, 2019.
A.Vijayasankar, Sharadha Kumar, M.A.Gopalan. On The Non-Homogeneous Quintic Equation with Five Unknowns , 10(8), 44-49, 2020.
A.Vijayasankar, Sharadha Kumar, M.A.Gopalan, On Non-Homogeneous Quinary Quintic Equation , South East Asian J. of Mathematics and Mathematical Sciences,18(1), 27-34, 2022.
N.Thiruniraiselvi, M.A.Gopalan, A Search on Integer Solutions to Ternary Non-homogeneous Nonic Diophantine Equation , "Advances in Nonlinear Variational Inequalities", 27(2), 13-17, 2024.
M.A.Gopalan, Sharadha Kumar,On Finding Integer Solutions on Binary Heptic Equation ,IRJEdT,8(7),782-787,2025.
M.A.Gopalan,Sharadha Kumar, Techniques on solving Binary Heptic Equation
,IJPREMS, 5(8), 91-95, 2025.
S.Devibala ,J.Shanthi,N.Thiruniraiselvi,M.A.Gopalan,Sharadhakumar,A Trek on the
Nonic Surface ,Bol.Soc.Paran.Mat.,43(2),1-5,2025
J.Shanthi, S.Devibala ,N.Thiruniraiselvi,M.A.Gopalan,Sharadhakumar, On Solving Non-
homogeneous Ternary Higher Degree Diophantine Equation
,Bol.Soc.Paran.Mat.,43(2),1-6,2025
N. Thiruniraiselvi, S. Devibala, J. Shanthi, M. A. Gopalan, Sharadha Kumar,
Patterns of Integer Solutions to Non-Homogeneous Ternary Sextic Diophantine
Equation A〖(x〗^2+y^2)-(2A-1)xy=(k^2+(4A-1) s^2)z^6
Bol.Soc.Paran.Mat.,43(3),1-7,2025
M.A.Gopalan,S.Vidhyalakshmi,J.Shanthi, On ternary cubic Diophantine equation
3(𝑥2+ 𝑦2) − 5𝑥𝑦 + 𝑥 + 𝑦 + 1 = 12𝑧3, International Journal of Applied Research, vol1(8), 209-212,2015.
M.A.Gopalan ,S.Vidhyalakshmi,J.Shanthi ,On Cubic Equation with Four Unknowns ,International Journal for Mathematics ,Vol 2(3),
Pg 1-8,2016.
M.A.Gopalan,S.Vidhyalakshmi,J.Shanthi ,On The Non-homogeneous Cubic Equation with Five Unknowns ,International Journal of Information Research and Review (IJIRR), Vol 3(6),Pg 2525-2528,2016.
M.A.Gopalan,J.Shanthi,On The Non-homogeneous Cubic Equation with Five Unknowns ,International Journal of Modern Sciences and Engineering Technology (IJMSET), Vol 3(5),Pg 32-36,2016.
E.Premalatha, J.Shanthi, M.A.Gopalan, On Non - Homogeneous Cubic Equation with Four Unknowns , Vol.14, Issue 5, 126-129, March 2021.
J.Shanthi,M.A.Gopalan, A search on Non –distinct Integer solutions to cubic Diophantine equation with four unknowns x^2-xy+y^2+4w^2=8z^3, International Research Journal of Education and Technology,(IRJEdT), Volume2,Issue01, 27- 32, May 2021.
S.Vidhyalakshmi, J.Shanthi, M.A.Gopalan, “On Homogeneous Cubic equation with four
Unknowns , International Journalof Engineering
Technology Research and Management, 5(7), 180-185, July 2021.
S.Vidhyalakshmi,J.Shanthi, M.A.Gopalan, T. Mahalakshmi, “On the non-homogeneous Ternary Cubic Diophantine equationw^2-z^2+2wx-2zx=x^3”, International Journal of Engineering Applied Science &Technology, Vol-7, Issue-3, 120-121, July-2022,.
M.A. Gopalan, J. Shanthi, V.Anbuvalli, Obervation on The Paper Entitled Solutions of the Homogeneous Cubic Eqution with Six Unknowns ,International Journal of Research Publication&Reviews, Vol-4, Issue-2, 313-317, Feb-2023.
J.Shanthi, S.Vidhyalakshmi,M.A.Gopalan, On Homogeneous Cubic Equation with Four
Unknowns(x^3+y^3 )=7zw^2,” Jananabha, Vol-53(1), 165-172, May-2023.
J. Shanthi , M.A. Gopalan, Cubic Diophantine equation of the form
Nxyz=w(xy+yz+zx) , International Journal of Modernization in Engineering Tech &Science, Vol-5, Issue-9, 1462-1463, Sep-2023.
J. Shanthi, M.A. Gopalan,”A Search on Integral Solutions to the Non- Homogeneous
Ternary Cubic Equationax^2+by^2=(a+b)z^3,a,b>0”, International Journal of Advanced Research in Science, Communication and Technology, Vol-4, Issue-1, 88-92, Nov-2024.
J.Shanthi,M.A.Gopalan,On finding Integer Solutions to Binary Cubic Equation ,International Journal of Multidisciplinary Research in Science, Engineering and Technology ,7(11), 16816-16820,2024.
J.Shanthi,M.A.Gopalan ,A Classification of Integer Solutions to Binary Cubic Equation ,International Journal of Progressive Research in Engineering Management and Science (IJPREMS) ,5(5),825-1828, 2025.
J.Shanthi,M.A.Gopalan,Observations on Binary Cubic Equation , IJARSCT, 5(1), 1-5, 2025.
J.Shanthi, M.A.Gopalan,On Solving Binary Cubic Equation , IRJEdT, 8(6), 139-144, 2025.
J.Shanthi, M.A.Gopalan, On finding Integer Solutions to Binary Quintic Equation , International Research Journal of Education andTechnology, 6(10), 226-231, 2024.
Sharadha Kumar, M.A.Gopalan, “On the Cubic Equation ”, JETIR, 6(1), 658-660, January 2019.
M.A.Gopalan, Sharadha Kumar, “On the non-homogeneous ternary cubic equation ”, International Journal of Engineering and Techniques, 4(5), 105-107, Sep-Oct (2018).
J.Shanthi, M.A.Gopalan, On finding Integer Solutions to Binary Quintic Equation , International Journal of Advanced Research in Science,Engineering and Technology, 11(10), 22369-22372, 2024.
J.Shanthi, M.A.Gopalan, Delineation of Integer Solutions to Non-Homogeneous QuinaryQuintic Diophantine Equation , International Journal of Research Publication and Reviews, 4(9), 1454-1457, 2023.
J.Shanthi,M.A.Gopalan, On the Ternary Non-homogeneous Quintic Equation , International Journal of Research Publication and Reviews, 4(8), 329-332, 2023.
J.Shanthi, M.A.Gopalan, On the Non-homogeneous Quinary Quintic Equation , International Research Journal of Education and Technology, 05 (08),238-245, 2023.
N.Thiruniraiselvi, M.A.Gopalan, The Non-Homogeneous Quintic Equation with Six Unknowns , International Journal for Research in Applied Science & Engineering Technology ,7(VI), 2527-2530, 2019.
A.Vijayasankar, Sharadha Kumar, M.A.Gopalan. On The Non-Homogeneous Quintic Equation with Five Unknowns , 10(8), 44-49, 2020.
A.Vijayasankar, Sharadha Kumar, M.A.Gopalan, On Non-Homogeneous Quinary Quintic Equation , South East Asian J. of Mathematics and Mathematical Sciences,18(1), 27-34, 2022.
N.Thiruniraiselvi, M.A.Gopalan, A Search on Integer Solutions to Ternary Non-homogeneous Nonic Diophantine Equation , "Advances in Nonlinear Variational Inequalities", 27(2), 13-17, 2024.
M.A.Gopalan, Sharadha Kumar,On Finding Integer Solutions on Binary Heptic Equation ,IRJEdT,8(7),782-787,2025.
M.A.Gopalan,Sharadha Kumar, Techniques on solving Binary Heptic Equation
,IJPREMS, 5(8), 91-95, 2025.
S.Devibala ,J.Shanthi,N.Thiruniraiselvi,M.A.Gopalan,Sharadhakumar,A Trek on the
Nonic Surface ,Bol.Soc.Paran.Mat.,43(2),1-5,2025
J.Shanthi, S.Devibala ,N.Thiruniraiselvi,M.A.Gopalan,Sharadhakumar, On Solving Non-
homogeneous Ternary Higher Degree Diophantine Equation
,Bol.Soc.Paran.Mat.,43(2),1-6,2025
N. Thiruniraiselvi, S. Devibala, J. Shanthi, M. A. Gopalan, Sharadha Kumar,
Patterns of Integer Solutions to Non-Homogeneous Ternary Sextic Diophantine
Equation A〖(x〗^2+y^2)-(2A-1)xy=(k^2+(4A-1) s^2)z^6
Bol.Soc.Paran.Mat.,43(3),1-7,2025
Published
2026-04-09
Section
Special Issue: Advances in Mathematical Sciences
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