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

1. Introduction

Number theory is one of the most fascinating and interesting subjects occupying an important place in the history of Mathematics. One of the interesting areas of Number Theory is the subject of Diophantine equations which has fascinated and motivated both Amateurs and Mathematicians alike. It is well-known that Diophantine equation is a polynomial equation in two or more unknowns requiring only integer solutions. The subject of Diophantine equations requiring only the integer solutions is an interesting area in the Theory of Numbers and it is the significant creation of the man-kind.

The beauty of Diophantine equations is that the number of equations is less than the number of unknowns. One can easily understand that the Diophantine problems are rich in variety playing a significant role in the development of Mathematics. The theory of Diophantine equations is popular in recent years providing a fertile ground for both Professionals and Amateurs. In addition to known results, this abounds with unsolved problems. Although many of its results can be stated in simple and elegant terms, their proofs are sometimes long and complicated.

There are unlimited varieties of third degree Diophantine equations which contribute in major part of research in this field.

[1, 2]

. We come across many problems in homogeneous or non-homogeneous cubic Diophantine equations with two or more variables. For the study of different choices of non-homogeneous cubic Diophantine equations with three unknowns by Gopalan et al, one may refer

[3-10]

. In

[11, 12]

, Vidhyalakshmi et al considered the non-homogeneous third degree equation for obtaining their integer solutions. For varieties of third degree Diophantine equations having four variables by Gopalan et al, look into the references mentioned in

[13-16]

. In

[17]

, Janaki and saranya considered a special cubic equation with four unknowns for its solutions in integers. Two more interesting cubic equations with four unknowns have been considered by Vidhyalakshmi et al

[18, 19]

. This paper concerns with getting infinitely many non-zero integral points for third degree Polynomial equation having two variables expressed as

. Substitution strategy is employed in obtaining successfully different choices of integral points to the above non-homogeneous third degree equation having two variables. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

Method of analysis

Consider the non-homogeneous third degree Diophantine equation with two unknowns represented by

(1)

Taking

(2)

in (1), it leads to

(3)

Let us write

(4)

which, after some calculations, is satisfied by

(5)

Assume the second solution to (4) as

(6)

where

is an unknown to be determined. Substituting (6) in (4) and simplifying, we have

and in view of (6), it is seen that

The repetition of the above process leads to the general solution to (4) as

(7)

From (3), we have

(8)

In view of (2), we have

(9)

Thus, (1) is satisfied by (9).

To characterize the nature of solutions and to obtain varieties of interesting relations among the solutions, one has to go for taking particular values to the parameters in (9).

2. Inspection-I

We take

in (1) and it is written as

(10)

The substitution of the linear transformations

(11)

in (10) leads to

(12)

Let

(13)

which, after some calculations, is satisfied by

(14)

Assume the second solution to (13) as

(15)

where

is to be found. Substituting (15) in (13) and simplifying, we have

and in view of (15), it is seen that

The repetition of the above process leads to the general solution to (13) as

(16)

From (12), we have

(17)

In view of (11), we have

(18)

Thus, (10) is satisfied by (18).

To obtain the relations among the solutions, one has to go for taking particular values to the parameters. For simplicity and brevity, we consider the integer solutions to (10) taking

in (18) and they are given by

(19)

A few numerical values for the obtained solutions (19) to equation (10) are shown in Table 1 as follows:

Table 1. Numerical values.

n	X₀(n)	y₀(n)
1	13²	11*13
2	2*24²	44*24
3	3*35²	99*35
4	4*46²	176*46
5	5*57²	275*57

From the above Table 1, it is seen that both the values of

are alternatively odd and even.

A few interesting relations among the integer solutions are presented below:

is a cubical integer

is a nasty number

is a perfect square

Procedure to obtain Ramanujan numbers of order Two:

The process of obtaining Ramanujan numbers of order Two from (10) is illustrated.

Illustration 1

Consider

From the above relation, one may observe that

Thus,

represents the second order Ramanujan number.

Illustration 2

Consider

In this case, the corresponding Second order Ramanujan number is found to be

Illustration 3

Consider

For this choice, the corresponding Second order Ramanujan number is found to be

Remark

It is worth mentioning that, apart from (14), other choice of integer solutions for (13) is given as

and taking

in (18), we have

3. Inspection-II

We take

in (1) and it is written as

(20)

Procedure 1

Taking

(21)

in (20), we have

(22)

which is satisfied by

(23)

Substituting (23) in (21), one has

(24)

Observe that (24) satisfies (20).

Some numerical examples for (20) are exhibited below in Table 2:

Table 2. Numerical examples.

n	x(n)	y(n)
1	16	12
2	98	84
3	300	270
4	676	624
5	1280	1200

Relations among the solutions:

is a cubical integer

is a multiple of 2

is a multiple of 4

represents the area of Pythagorean triangle

is a perfect square for

10.

11.

is a cubical integer

12.

13.

is written as the difference of two squares

14.

15.

16.

is a perfect square when

takes the values

17.

18.

19.

Procedure to obtain Ramanujan numbers of order Two:

The process of obtaining Ramanujan numbers of order Two from (20) is illustrated.

Illustration 4

From the above relation, one may observe that

Thus,

represents the second order Ramanujan number.

Illustration 5

Consider

In this case, the corresponding Second order Ramanujan number is found to be

Illustration 6

Consider

From the above relation, one may observe that

Thus,

represents the second order Ramanujan number.

Remark:

In addition to the solutions (24), we have an another set of solutions to (20) given by

4. Conclusions

In this article, the substitution strategy is utilized to obtain successfully integer solutions for third degree Polynomial equation having two variables. The readers may search for different approaches to analyze third degree equations with two unknowns. Further, they may search for varieties of relations through the obtained integer solutions.

Notations

Author Contributions

Nagarajan Thiruniraiselvi: Data curation, Investigation, Methodology, Resources, Visualization, Writing – review & editing

Sharadha Kumar: Formal Analysis, Methodology, Project administration, Validation, Writing – original draft

Mayilrangam Ambravaneswaran Gopalan: Conceptualization, Methodology, Resources, Supervision, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

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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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
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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Author Information

Nagarajan Thiruniraiselvi

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

Biography: Nagarajan Thiruniraiselvi is working as Assistant Professor in Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Trichy-621 112 and is teaching Mathematics since 06 years. She is interested in finding solutions to different types of Diophantine Equation and Number patterns. She has published more than 75 papers in National and International journals. She, along with her colleagues, has published 03 books in the area of Diophantine equations and presented 15 papers in various International Conferences. She is a reviewer for “Journal of Experimental Agriculture International” and “Asian Research Journal of Mathematics

Research Fields: Number Theory

Contact Email

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

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

Biography: Sharadha Kumar was lecturer in in Department of Mathematics MES College, Bangalore. She got Doctor of Philosophy in Number Theory and her area of interest is solving Diophantine equations in various disciplines. She has published 30 papers in National and International journals and is a co-author for one published book and presented 12 papers in various International Conferences.

Research Fields: Number Theory

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

Biography: Mayilrangam Ambravaneswaran Gopalan is currently Professor of Mathematics at Shrimati Indira Gandhi College, Tiruchirappalli and has taught Mathematics for nearly three decades. He is interested in problem solving in the area of Diophantine equations and Number patterns. He has published more than 700 papers in National and International journals. He, along with his colleagues, has published 13 books in the area of Diophantine equations and Number patterns. He serves on the editorial boards of IJPMS and IJAR and a life member of Kerala Mathematics Association.

Research Fields: Number Theory

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

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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}
}

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  -

Copy | Download