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.
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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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Article Publishing History
Received: | 12-02-2024 |
---|---|
Accepted: | 02-04-2024 |
Reviewed by: | Sonendra Kumar Gupta |
Second Review by: | 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 integerk (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.
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