A Study on two Curious Diophantine Problems
M.A.Gopalan α, K.Meena σ & S. Vidhyalakshmi ρ
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ABSTRACT
In this communication, an attempt has been made to obtain pairs of nonzero distinct integers such that, in each pair,
 The square of the sum added with the cube of the sum is equal to two times the sum of the cubes of the corresponding integers.
 2k times the square of the sum added with the cube of the sum is equal to two times the sum of the cubes of the corresponding integers.
Keywords: diophantine problems, binary cubic, integer solutions.
Author α : Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy620 002, Tamil Nadu, India.
σ: Former VC, Bharathidasan University, Trichy620 024, Tamil Nadu, India.
 INTRODUCTION
The subject of diophantine equations is one of the major areas in the history of Number Theory. It is obviously a broad topic and has a marvelous effect on credulous people and always occupy a remarkable position due to unquestioned historical importance because of the limitless supply of exciting nonroutine and challenging problems [14]. All that needed is something to arouse interest and get started.
In this communication, an attempt has been made to obtain pairs of nonzero distinct integers such that, in each pair,
It is worth mentioning that in case (i), the process employs the integer solutions of the pellian equation and in case (ii), the integer solutions of the positive pell equation are employed.
 METHOD OF ANALYSIS
Problem: 1
Let be two distinct nonzero integers. The problem under consideration is mathematically equivalent to solving the binary cubic Diophantine equation represented by
(1)
Introduction of linear transformations
(2)
in (1) leads to the positive pellian equation
which is satisfied by
where
In view of (2), the values of and satisfying (1) are given by
A few numerical examples are exhibited in Table: 1 below:
Table 1: Numerical examples
  
0  5  1 
1  76  20 
2  1065  285 
3  14840  3976 
The recurrence relations satisfied by, are respectively presented below:
Observation 1:
From each of the values of , , one may generate second order Ramanujan Numbers.
Illustration:
Consider
Now,
Thus, 294905, 284873, 12665 represent second order Ramanujan numbers
Observation: 2
It is seen that
say
The pair is a Diophantine 2tuple with property
Now,
Note that
The above relations lead to the result that the triple is a Diophantine 3tuple with property
Remark:
It is to be noted that, by considering either the pair or , one generates two more Diophantine 3tuples with property . Thus, the repeated applications of the above process leads to the generation of sequence of diophantine 3tuples with property .
Problem: 2
We search for two distinct nonzero integers a, b such that
(3)
Introduction of linear transformations
(4)
in (3) leads to the positive pellian equation
(5)
where
(6)
The fundamental solution of (5) is
To obtain the other solutions of (5), consider the pell equation
whose solution is given by
where
Applying Brahmagupta lemma between and, the other integer solutions of (5) are given by
From (6), we get
In view of (4), the values of and satisfying (3) are given by
Note:
For properties, one has to go for particular values for k
Illustration:
Let. Then the corresponding values of a, b are given by
A few numerical examples are exhibited in Table: 2 below:
Table 2: Numerical examples



0  2  0 
1  10  2 
2  40  10 
3  152  40 
4  570  152 
The recurrence relations satisfied by are respectively presented below:
Observation: 3
From each of the values of , one may generate second order Ramanujan Numbers.
Illustration:
Consider
Thus, 130, 1810 represent second order Ramanujan numbers
Observation: 4
It is seen that
say
The pair is a Diophantine 2tuple with property
Now,
Note that
The above relations lead to the result that the triple is a Diophantine 3tuple with property
Following the remark presented in problem: 1, in this case also, a sequence of diophantine 3tuples is generated with property
In conclusion, the readers of this communication may search for other formulations of Diophantine problems involving different pellian equations.
REFERENCES
 J.N. Kapur, Fascinating world of Mathematical Sciences, Vol.14, Mathematical Sciences Trust Society, New Delhi, 1994.
 Shailesh Shirali, Mathematical Marvels, A primes on number sequences, Universities Press, India, 2001.
 Titu Andreescu, Dorin Andrica and Zuming Feng, 104 Number Theory problems, Birkhauser Boston Inc., 2007.
 Titu Andreescu and Dorin Andrica, Number Theory, Birkhauser Boston Inc., 2009.