Non-Homogeneous Binary Cubic Equation a(x-y)<sup>3</sup>=8bxy

Nagarajan Thiruniraiselvi; Sharadha Kumar; Mayilrangam Ambravaneswaran Gopalan

doi:doi:10.11648/j.pamj.20241302.12

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Non-Homogeneous Binary Cubic Equation a(x-y)³=8bxy

Nagarajan Thiruniraiselvi^*

, Sharadha Kumar

, Mayilrangam Ambravaneswaran Gopalan

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 2)

Received: 21 May 2024 Accepted: 7 June 2024 Published: 19 June 2024

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Abstract

Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)³=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

Published in	Pure and Applied Mathematics Journal (Volume 13, Issue 2)
DOI	10.11648/j.pamj.20241302.12
Page(s)	29-35
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Non-Homogeneous Cubic, Binary Cubic, Integer Solutions, Ramanujan Numbers

References

[1]	L. E. Dickson, History of Theory of Numbers, Vol. 2, Chelsea Publishing Company, NewYork, 1952.
[2]	L. J. Mordell, Diophantine equations, Academic press, New York, 1969.
[3]	M. A. Gopalan, G. Sangeetha, “On the ternary cubic Diophantine equation y²=Dx²+z³”, Archimedes J. Math 1(1), 2011, 7-14.
[4]	M. A. Gopalan, B. Sivakami, “Integral solutions of the ternary cubic equation 4x²-4xy+6y²=((k+1)²+5)w³”, Impact J. Sci. Tech, 6(1), 2012, 15-22.
[5]	M. A. Gopalan, B. Sivakami, “On the ternary cubic Diophantine equation 2xz=y²(x+z)”, Bessel J. Math, 2(3), 2012, 171-177.
[6]	M. A. Gopalan, K. Geetha, “On the ternary cubic Diophantine equation x²+y²-xy=z³”, Bessel J. Math., 3(2), 2013, 119-123.
[7]	M. A. Gopalan, S. Vidhyalakshmi, A. Kavitha “Observations on the ternary cubic equation x²+y²+xy=12z³”, Antartica J. Math, 10(5), 2013, 453-460.
[8]	M. A. Gopalan, S. Vidhyalakshmi, K. Lakshmi, “Lattice points on the non-homogeneous cubic equation x³+y³+z³+(x+y+z)=0”, Impact J. Sci. Tech, 7(1), 2013, 21-25.
[9]	M. A. Gopalan, S. Vidhyalakshmi, K. Lakshmi “Lattice points on the non-homogeneous cubic equation x³+y³+z³-(x+y+z)=0”, Impact J. Sci. Tech, 7(1), 2013, 51-55.
[10]	M. A. Gopalan, S. Vidhyalakshmi, S. Mallika, “On the ternary non-homogenous cubic equation x³+y³-3(x+y)=2(3k²-2)z³”, Impact J. Sci. Tech, 7(1), 2013, 41-45.
[11]	S. Vidyalakshmi, T. R. Usharani, M. A. Gopalan, “Integral solutions of non-homogeneous ternary cubic equation ax²+by²=(a+b)z³”, Diophantus J. Math 2(1), 2013, 31-38.
[12]	S. Vidhyalakshmi, M. A. Gopalan, S. Aarthy Thangam, “On the ternary cubic Diophantine equation 4(x²+x)+5(y²+2y)=-6+14z³” International Journal of Innovative Research and Review (JIRR), 2(3), July-Sep 2014, 34-39.
[13]	M. A. Gopalan, S. Vidhyalakshmi, G. Sumathi, “On the homogeneous cubic equation with four unknowns X³+Y³=14Z³-3W²(X+Y)”, Discovery, 2(4), 2012, 17-19.
[14]	M. A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, “On homogeneous cubic equation with four unknowns (x+y+z)³=z(xy+31w²)”, Cayley J. Math, 2(2), 2013, 163-168.
[15]	M. A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, “On homogeneous cubic equation with four unknowns x³+y³=21zw²”, Review of Information Engineering and Applications, 1(4), 2014, 93-101.
[16]	M. A. Gopalan, S. Vidhyalakshmi, E. Premalatha, C. Nithya, “On the cubic equation with four unknowns x³+y³=31(k²+3s²)zw²”, IJSIMR, 2(11), Nov-2014, 923-926.
[17]	G. Janaki, C. Saranya, “Integral solutions of the ternary cubic equation 3(x²+y²)-4xy+2(x+y+1)=972z³”, IRJET, 4(3), March 2017, 665-669.
[18]	S. Vidhyalakshmi, M. A. Gopalan, A. Kavitha, “Observation on homogeneous cubic equation with four unknowns X³+Y³=7²ⁿZW²”, IJMER, 3(3), May-June 2013, 1487-1492.
[19]	S. Vidhyalakshmi, T. Mahalakshmi and M. A. Gopalan, “A Search On The Integer Solutions of Cubic Diophantine Equation with Four Unknowns x²+y²+4(35z²-4-w²)=6xyz”, International Journal of Grid and Distributed Computing, 13(2021), 2, 2581-2584.

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APA Style

Thiruniraiselvi, N., Kumar, S., Gopalan, M. A. (2024). Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure and Applied Mathematics Journal, 13(2), 29-35. https://doi.org/10.11648/j.pamj.20241302.12

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ACS Style

Thiruniraiselvi, N.; Kumar, S.; Gopalan, M. A. Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure Appl. Math. J. 2024, 13(2), 29-35. doi: 10.11648/j.pamj.20241302.12

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AMA Style

Thiruniraiselvi N, Kumar S, Gopalan MA. Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure Appl Math J. 2024;13(2):29-35. doi: 10.11648/j.pamj.20241302.12

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@article{10.11648/j.pamj.20241302.12,
  author = {Nagarajan Thiruniraiselvi and Sharadha Kumar and Mayilrangam Ambravaneswaran Gopalan},
  title = {Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {2},
  pages = {29-35},
  doi = {10.11648/j.pamj.20241302.12},
  url = {https://doi.org/10.11648/j.pamj.20241302.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241302.12},
  abstract = {Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)3=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.
},
 year = {2024}
}

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TY  - JOUR
T1  - Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy

AU  - Nagarajan Thiruniraiselvi
AU  - Sharadha Kumar
AU  - Mayilrangam Ambravaneswaran Gopalan
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PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241302.12
DO  - 10.11648/j.pamj.20241302.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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EP  - 35
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241302.12
AB  - Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)3=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

VL  - 13
IS  - 2
ER  -

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Author Information

Nagarajan Thiruniraiselvi

Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Samayapuram, Trichy, India

Contact Email

http://orcid.org/0000-0003-4652-3846
Sharadha Kumar

Department of Mathematics, National College, Affiliated to Bharathidasan University, Trichy, India

Contact Email

http://orcid.org/0000-0002-0509-6158
Mayilrangam Ambravaneswaran Gopalan

Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy, India

Contact Email

http://orcid.org/0000-0003-1307-2348

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Plain Text BibTeX RIS

APA Style

Thiruniraiselvi, N., Kumar, S., Gopalan, M. A. (2024). Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure and Applied Mathematics Journal, 13(2), 29-35. https://doi.org/10.11648/j.pamj.20241302.12

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ACS Style

Thiruniraiselvi, N.; Kumar, S.; Gopalan, M. A. Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure Appl. Math. J. 2024, 13(2), 29-35. doi: 10.11648/j.pamj.20241302.12

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AMA Style

Thiruniraiselvi N, Kumar S, Gopalan MA. Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy. Pure Appl Math J. 2024;13(2):29-35. doi: 10.11648/j.pamj.20241302.12

Copy | Download

@article{10.11648/j.pamj.20241302.12,
  author = {Nagarajan Thiruniraiselvi and Sharadha Kumar and Mayilrangam Ambravaneswaran Gopalan},
  title = {Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {2},
  pages = {29-35},
  doi = {10.11648/j.pamj.20241302.12},
  url = {https://doi.org/10.11648/j.pamj.20241302.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241302.12},
  abstract = {Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)3=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Non-Homogeneous Binary Cubic Equation a(x-y)3=8bxy

AU  - Nagarajan Thiruniraiselvi
AU  - Sharadha Kumar
AU  - Mayilrangam Ambravaneswaran Gopalan
Y1  - 2024/06/19
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241302.12
DO  - 10.11648/j.pamj.20241302.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 29
EP  - 35
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241302.12
AB  - Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)3=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

VL  - 13
IS  - 2
ER  -

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