A Study on Two Curious Diophantine Problems

London Journal of Research in Science: Natural and Formal
Volume | Issue | Compilation
Authored by M.A. Gopalan , K. Meena, S. Vidhyalakshmi
Classification: FOR Code: 11D99
Keywords: diophantine problems, binary cubic, integer solutions.
Language: English

In this communication, an attempt has been made to obtain pairs of non-zero distinct integers such that, in each pair,

i) 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.

ii) 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.

               

A Study on two Curious Diophantine Problems

M.A.Gopalan α,  K.Meena σ  &  S. Vidhyalakshmi ρ

____________________________________________

ABSTRACT

In this communication, an attempt has been made to obtain pairs of non-zero distinct integers such that, in each pair,  

  1. 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.
  2. 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, Trichy-620 002, Tamil Nadu, India.

σ: Former VC, Bharathidasan University, Trichy-620 024, Tamil Nadu, India.        

  1. 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 non-routine and challenging problems [1-4]. All that needed is something to arouse interest and get started.

In this communication, an attempt has been made to obtain pairs of non-zero 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.

  1. METHOD OF ANALYSIS

Problem: 1

Let  be two distinct non-zero 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 2-tuple with property

Now,

Note that

The above relations lead to the result that the triple  is a Diophantine 3-tuple with property

Remark:

It is to be noted that, by considering either the pair  or , one generates two more Diophantine 3-tuples with property . Thus, the repeated applications of the above process leads to the generation of sequence of diophantine 3-tuples with property .

Problem: 2

We search for two distinct non-zero 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 2-tuple with property

Now,

Note that

The above relations lead to the result that the triple  is a Diophantine 3-tuple with property

Following the remark presented in problem: 1, in this case also, a sequence of diophantine 3-tuples is generated with property

In conclusion, the readers of this communication may search for other formulations of Diophantine problems involving different pellian equations.

REFERENCES

  1. J.N. Kapur, Fascinating world of Mathematical Sciences, Vol.14, Mathematical Sciences Trust Society, New Delhi, 1994.
  2. Shailesh Shirali, Mathematical Marvels, A primes on number sequences, Universities Press, India, 2001.
  3. Titu Andreescu, Dorin Andrica and Zuming Feng, 104 Number Theory problems, Birkhauser Boston Inc., 2007.
  4. Titu Andreescu and Dorin Andrica, Number Theory, Birkhauser Boston Inc., 2009.



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