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Binary Third degree Diophantine Equation 5 (x-y)3 = 8xy

N Thiruniraiselvi1* and M A Gopalan2

1Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Samayapuram, Trichy, Tamil Nadu India .

2Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy, Tamil Nadu India .

Corresponding author Email: drntsmaths@gmail.com

This article emphasizes on finding non-zero different integer solutions to binary third degree Diophantine equation 5 (x-y)3 = 8xy . Two different sets of solutions in integers are presented. Some fascinating relations from the solutions are obtained. The method to get second order Ramanujan   numbers is illustrated.


Binary cubic; Integer Solutions; Non-homogeneous cubic; Ramanujan numbers

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Thiruniraiselvi N, Gopalan M. A. Binary Third degree Diophantine Equation 5 (x-y)3 = 8xy. Oriental Jornal of Physical Sciences 2024; 9(1).

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Thiruniraiselvi N, Gopalan M. A. Binary Third degree Diophantine Equation 5 (x-y)3 = 8xy. Oriental Jornal of Physical Sciences 2024; 9(1). Available here:https://bit.ly/3VCQxCi

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Publish History

Article Publishing History

Received: 12-02-2024
Accepted: 02-04-2024
Reviewed by: Orcid Orcid Sonendra Kumar Gupta
Second Review by: Orcid Orcid Dr. Jaime Rodriguez
Final Approval by: Dr. Amit Kumar Verma

Introduction

The third degree Diophantine equations are enormous in variety and they have contributed to expansion of research in this field[1,2]. For an extensive approach of these types of problems , one may refer [3-28]. In this article a search is made to get solutions in integers for the considered problem through employing linear transformations. Also , the method of getting second order  Ramanujan   numbers from the obtained solution is discussed. Some fascinating relations from the solutions are presented.

Method of analysis

The  non-homogeneous binary third degree equation under consideration is

The substitution of the linear transformations

  in (1) leads to

Let

which , after some calculations , is satisfied by    

      

Assume the second solution to (4) as

where h 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

From (3) ,we have

                

 In view of (2) , we have

                                                  

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

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 (1) taking          

    

in (9) and they are given by

A few numerical values for the obtained solutions (9) to equation (1) are presented in Table 1 below:

Table 1: Numerical values

x(n)

y(n)

1

49

35

2

288

240

3

3*289

17*45

4

4*484

22*80

5

5*729

27*125

From the above Table 1, it is seen that both the values of x(n), y(n) are alternatively odd and even.

A few interesting relations among the solutions are presented below:

5[2y(k) – x(k) + 4k] is a cubical integer

k (2 y(k) – x(k)) is written as difference of two squares

k (2 x(k) – y(k)) is written as difference of two squares

x(k) – y(K) – 2Ct6, k + 2k + 2 is a perfect square

x(k) – y(k) – 13k =t22.k



25 x y is a cubical integer

25k3 x(k) = (y(k))2

x(2n) – y(2n) = Th2n + Mn + 2 + 7M2n + 9

x(k) – y(k) is a perfect square when k takes the values



where

[x(k + 2) – y(k +2)] – 2[x(k +1) – y(k + 1)]+[x(k) – y(k)] = 20

[x(k + 4) – y(k +4)] – 2[x(k +3) – y(k + 3)] + 2[x(k + 2) – y(k + 2)] – 2[x(k + 1) – y(k +1)] + [x(k) – y(k)] = 40

[x(n + k) – y(n + k)] – [x(n + k - 1) – y(n + k - 1)] = 20 (k + n) -6

y(k) = 20 Pk5 + 6CPk15 +9k

y(k) = 20 Pk5 + 3CPk16 + 3CPk14 + 9k

Formulation of Second order Ramanujan numbers

From each of  the solutions of (1) given by (9) , one can find Second order Ramanujan numbers with base numbers as real integers .

Illustration 1

Consider

From the above relation , one may observe that

Thus , 50k4 + 20k3 + 54k2 + 20k + 4 represents the second order Ramanujan number.

Illustration 2

Consider

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

It is worth mentioning that , in addition to the solutions (5) , we have an another set of solutions in integers to (4)  given by

and taking          

in (9)  , the corresponding integer solutions to (1) are given by

Conclusion

This article gives an approach to solve third degree equation with two unknowns though different methods to get solutions in integers. The research in this field may attempt to find various other methods to solve binary cubic equation and also approach to get second order Ramanujan numbers and find various other relation from the obtained solution.     

Acknowledgment

The authors are grateful to the reviewers for their comments and guidance .

Funding Sources

The authors have no financial support for the research and publication of this article.

Conflict of Interest

No conflict of interest with anyone.

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, S. Vidhyalakshmi, G. Sumathi, “On the homogeneous cubic equation with four unknowns X3 + Y3 = 14Z3 – 3W2(X + Y)” ”, Discovery, 2(4), 2012, 17-19.
  7. S. Vidhyalakshmi, M.A. Gopalan, A. Kavitha, “Observation on homogeneous cubic equation with four unknowns X3 + Y3 = 72n ZW2 ”, IJMER, 3(3), May-June 2013, 1487-1492.
  8. 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.
  9. M.A. Gopalan, K. Geetha, “On the ternary cubic Diophantine equation x2 + y2 – xy = z3  ”, Bessel J.Math., 3(2), 2013,119-123.
  10. 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.
  11. 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.
  12. 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,
  13. 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.
    CrossRef
  14. M.A. Gopalan, S. Vidhyalakshmi, N. Thiruniraiselvi, “On homogeneous cubic equation with four unknowns (x + y + z)3 = z(xy + 3lw2) ”, 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 – 2lzw2 ”, Review of Information Engineering and Applications, 1(4), 2014, 93-101.
    CrossRef
  16. 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.
  17. M.A. Gopalan, S. Vidhyalakshmi, E. Premalatha, C. Nithya, “On the cubic equation with four unknowns x3 + y3 = 3l(k2 +3s2)zw2 ”, IJSIMR, 2(11), Nov-2014, 923-926.
    CrossRef
  18. S. Vidhyalakshmi, T.R. Usha Rani, M.A. Gopalan, V. Kiruthika, “On the cubic equation with four unknowns x3 + y3 = 14zw2 ”, IJSRP, 5(3), March 2015, 1-11.
  19. M.A. Gopalan, S. Vidhyalakshmi, J. Shanthi, “On the cubic equation with four unknowns x3 + 4z3 = y3 + 4w3 + 6(x - y)2 ”, International Journal of Mathematics Trends and Technology, 20(1), April 2015, 75-84.
    CrossRef
  20. M.A. Gopalan, N. Thiruniraiselvi and V. Kiruthika, “On the ternary cubic diophantine equation 7x2 – 4y2 = 3z3 ”, IJRSR, 6(9), Sep-2015, 6197-6199.
  21. M.A. Gopalan, S. Vidhyalakshmi, J. Shanthi, J. Maheswari, “On ternary cubic diophantine equation 3(x2 + y2) – 5xy + x + y + 1= 12z3 ”, International Journal of Applied Research, 1(8), 2015, 209-212.
  22. R. Anbuselvi, K. Kannaki, “On ternary cubic diophantine equation 3(x2 + y2) – 5xy + x + y + 1= 15z3 ”, IJSR, 5(9), Sep 2016, 369-375.
    CrossRef
  23. 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.
  24. R. Anbuselvi, K.S. Araththi, “On the cubic equation with four unknowns x3 + y3 = 24zw2 ”, IJERA, 7(11) (Part-I), Nov-2017, 01-06.
    CrossRef
  25. Sharadha Kumar, M.A. Gopalan, ON THE CUBIC EQUATION x3 + y3 + 6(x + y) z2 = 4w3 ", JETIR ,6(1), 2019, 658-660.
  26. S. Vidhyalakshmi, T. Mahalakshmi, M.A. Gopalan, “A Search On the Integer Solutions of Cubic Diophantine Equation with Four Unknowns x3 – y3 = 4(w3 = z3) + 3 (x - y)3 ”, International Journal of Engineering And Science, 10(8), August 2020, 13-17.
  27. S. Vidhyalakshmi, T. Mahalakshmi, M.A.Gopalan and G. Dhanalakshmi, “On homogeneous Cubic Equation with Four Unknowns (x3 + y3) = 7zw2 ” Stochastic Modeling & Applications, 25(3), Special Issue III, Part-3, July-December 2021, 1968-1978.
  28. 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(2), 2021, 2581-2584.
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