On Binary Diophantine Equation 3x2 5y2 = 12

On Binary Diophantine Equation

3

x

2



5

y

2



12

S. Mallika

1

, V. Surya

2 1

Assistant Professor, 2M. Phil Scholar, Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India.

Abstract: Non-homogeneous binary quadratic equation representing hyperbola given by

3

x

2



5

y

2



12

_{is analyzed for its} non-zero distinct integer solutions. A few interesting relations among its solutions are presented. Also, knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbola and parabola are presented.

Keywords: Binary quadratic, Hyperbola, Parabola, Integral solutions, Pell equation. AMS Mathematics subject Classification (2010):11D0

I. INTRODUCTION

The binary quadratic Diophantine equations of the form

ax

2



by

2



N

,

(

a

,

b

,

c



0

)

are rich in variety and have been analyzed by many mathematicians for their respective integer solutions for particular values of

a

,

b

and

N

.

In this context, one may refer [1-18].

This communication concerns with the problem of obtaining non-zero distinct integer solutions to the binary quadratic equation

given by,

3

x

2



5

y

2



12

representing hyperbola. A few interesting relations among its solutions are presented. Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented.

II. METHOD OF ANALYSIS

The Diophantine equation representing the binary quadratic equation to be solved for its non-zero distinct integral solution is

3

x

2



5

y

2



12

(1) Introduce the linear transformation

x



X



5

T

,

y



X



3

T

(2) From (1) & (2) we have,

X

2



15

T

2



6

(3) whose smallest positive integer solution is

1

,

3

₀

0



T



X

To obtain the other solutions of (3), consider the Pell equation

X

2



15

T

2



1

(4) whose smallest positive integer solution is

1

~

,

4

~

0 0



T



X

whose general solution is given by

n n

n

n

g

X

f

T

2

1

~

,

15

2

1

~





where





1





1

15

4

15

4













n n

n

f





1





1

15

4

15

4













n n

n

g

Applying Brahmagupta lemma between



x

0

,

y

0



and



x

n

y

n



~

,

~

, the other integer solutions of (1) are given by

n n

n

f

g

x

4

15

15

15

_1





n n

n

f

g

y

3

15

12

The recurrence relations satisfied by x and y are given by

0

8

_{2 1}

3











 n n

n

x

x

x

0

8

_{2 1}

3











 n n

n

y

y

y

Some numerical examples of x and y satisfying (1) are given in the Table: 1 below:

Table: 1 Numerical examples

n

1

 n

x

y

_n1

-1 8 6

0 62 48

1 488 378

2 3842 2976

3 30248 23430

4 288142 184464

From the above table, we observe some interesting relations among the solutions which are presented below:

A.

x

n1 _&

y

n1_{values are always even .}

B. Relations among the solutions

1)

x

_n3



8

x

_n2



x

_n1



0

2)

5

y

n1



x

n2



4

x

n1



0

3)

15

y

n3



93

x

n2



12

x

n1



0

4)

40

y

_n3



x

_n1



3

x

_n3



0

5)

9

x

_n2



36

x

_n1



45

y

_n1



0

6)

x

_n3



31

x

_n1



40

y

_n1



0

7)

y

_n3



24

x

_n1



31

y

_n1



0

8)

x

_n3



x

_n1



10

y

_n2



0

9)

4

y

_n1



3

x

_n1



y

_n2



0

10)

3

x

_n1



24

x

_n2



3

x

_n3



0

11)

5

y

_n1



31

x

_n2



4

x

_n3



0

12)

5

y

_n2



4

x

_n2



x

_n3



0

13)

y

_n3



6

x

_n2



y

_n1



0

14)

x

_n1



4

x

_n2



5

y

_n2



0

15)

y

_n1



3

x

_n2



4

y

_n2



0

16)

4

x

_n1



31

x

_n2



5

y

_n3



0

17)

4

y

_n2



3

x

_n2



y

_n3



0

18)

4

y

_n1



3

x

_n3



31

y

_n2



0

19)

x

_n1



31

x

_n3



40

y

_n3



0

21)

y

_n2



3

x

_n3



4

y

_n3



0

22)

24

x

_n3



y

_n1



31

y

_n3



0

23)

8

y

n2



y

n1



y

n3



0

24)

3

x

_n1



31

y

_n2



4

y

_n3



0

C. Each of the following expressions represents a nasty number

1)



48

x

2n2



6

x

2n3



12



2)



378

6

96



8

1

4 2 2

2n



x

n



x

3)



24

x

_2n2



30

y

_2n2



12



4)



558

90

144



12

1

3 2 2

2n



y

n



x

5)



1464

30

372



31

1

4 2 2

2n



y

n



x

6)



1134

144

36



3

1

4 2 3

2n



x

n



x

7)



6

x

_2n3



60

y

_2n2



12



8)



186

x

2n3



240

y

2n3



12



9)



366

x

_2n3



60

y

_2n4



12



10)



24

1890

372



31

1

2 2 4

2n



y

n



x

11)



186

1890

48



4

1

3 2 4

2n



y

n



x

12)



1464

x

_2n4



1890

y

_2n4



12



13)



24

186

36



3

1

2 2 3

2n



y

n



y

14)



6

366

72



6

1

2 2 4

2n



y

n



y

15)



186

1464

36



3

1

3 2 4

2n



y

n



y

D. Each of the following expressions represents a cubical integer

1)



8

x

3n3



x

3n4



24

x

n1



3

x

n2



2)



63

_{3 3 3 5}

189

₁

3

₃



8

1

 







n



n



n

n

x

x

x

x

3)



4

x

_3n3



5

y

_3n3



12

x

_n1



15

y

_n1



4)



31

_{3 3}

5

_{3 4}

93

₁

15

₂



4

1

 







n



n



n

n

y

x

y

5)



244

_{3 3}

5

_{3 5}

732

₁

15

₃



31

1

  





n



n



n

n

y

x

y

x

6)



63

x

_3n4



8

x

_3n5



189

x

_n2



24

x

_n3



7)



x

_3n4



10

y

_3n3



3

x

_n2



30

y

_n1



8)



31

x

_3n4



40

y

_3n4



93

x

_n2



120

y

_n2



9)



61

x

_3n4



10

y

_3n5



183

x

_n2



30

y

_n3



10)



4

_{3 5}

315

_{3 3}

12

₃

945

₁



31

1

 







n



n



n

n

y

x

y

x

11)



31

_{3 5}

315

_{3 4}

93

₃

945

₂



4

1

 







n



n



n

n

y

x

y

x

12)



244

x

_3n5



315

y

_3n5



732

x

_n3



945

y

_n3



13)



4

_{3 4}

31

_{3 3}

12

₂

93

₁



3

1

 







n



n



n

n

y

y

y

y

14)



_{3 5}

61

_{3 3}

3

₃

183

₁



6

1

 







n



n



n

n

y

y

y

y

15)



31

_{3 5}

244

_{3 4}

93

₃

732

₂



3

1

 







n



n



n

n

y

y

y

y

E. Each of the following expressions represents a bi-quadratic integer

1)



8

x

_4n4



x

_4n5



32

x

_2n2



4

x

_2n3



6



2)



63

252

4

48



8

1

4 2 2 2 6 4 4

4n



x

n



x

n



x

n



x

3)



4

x

4n4



5

y

4n4



16

x

2n2



20

y

2n2



6



4)



31

5

124

20

24



4

1

3 2 2 2 5 4 4

4n



y

n



x

n



y

n



x

5)



244

5

976

20

186



31

1

4 2 2 2 6 4 4

4n



y

n



x

n



y

n



x

6)



63

x

_4n5



8

x

_4n6



252

x

_2n3



32

x

_2n4



6



7)



x

_4n5



10

y

_4n4



4

x

_2n3



40

y

_2n2



6



8)



31

x

_4n5



40

y

_4n5



124

x

_2n3



160

y

_2n3



6



9)



61

x

4n5



10

y

4n6



244

x

2n3



40

y

2n4



16



10)



4

315

16

1260

186



31

1

2 2 4 2 4 4 6

4n



y

n



x

n



y

n



x

11)



31

315

124

1260

24



4

1

3 2 4 2 5 4 6

4n



y

n



x

n



y

n



x

12)



244

x

_4n6



315

y

_4n6



976

x

_2n4



1260

y

_2n4



6



13)



4

31

16

124

18



3

1

2 2 3 2 4 4 5

4n



y

n



y

n



y

n



14)



61

4

244

36



6

1

2 2 4

2 4 4 6

4n



y

n



y

n



y

n



y

15)



31

244

124

976

18



3

1

3 2 4

2 5

4 6

4n



y

n



y

n



y

n



y

F. Each of the following expressions represents a quintic integer

1)



8

x

_5n5



x

_5n6



40

x

_3n3



5

x

_3n4



80

x

_n1



10

x

_n2



2)



63

_{5 5 5 7}

315

_{3 3}

5

_{3 5}

630

₁

10

₃



8

1

 

 







n



n



n



n



n

n

x

x

x

x

x

x

3)



4

x

5n5



5

y

5n5



20

x

3n3



25

y

3n3



40

x

n1



50

y

n1



4)



31

_{5 5}

5

_{5 6}

155

_{3 3}

25

_{3 4}

310

₁

50

₂



4

1

 

 







n



n



n



n



n

n

y

x

y

x

y

x

5)



244

_{5 5}

5

_{5 7}

1220

_{3 3}

25

_{3 5}

2440

₁

50

₃



31

1

 

 







n



n



n



n



n

n

y

x

y

x

y

x

6)



63

x

_5n6



8

x

_5n7



315

x

_3n4



40

x

_3n5



630

x

_n2



80

x

_n3



7)



x

5n6



10

y

5n5



5

x

3n4



50

y

3n3



10

x

n2



100

y

n1



8)



31

x

_5n6



40

y

_5n6



155

x

_3n4



200

y

_3n4



310

x

_n2



400

y

_n2



9)



61

x

_5n6



10

y

_5n7



305

x

_3n4



50

y

_3n5



610

x

_n2



100

y

_n3



10)



4

_{5 7}

315

_{5 5}

20

_{3 5}

1575

_{3 3}

40

₃

3150

₁



31

1

 

 







n



n



n



n



n

n

y

x

y

x

y

x

11)



31

_{5 7}

315

_{5 6}

155

_{3 5}

1575

_{3 4}

310

₃

3150

₂



4

1

 

 







n



n



n



n



n

n

y

x

y

x

y

x

12)



244

x

_5n7



315

y

_5n7



1220

x

_3n5



1575

y

_3n5



2440

x

_n3



3150

y

_n3



13)



4

_{5 6}

31

_{5 5}

20

_{3 4}

155

_{3 3}

40

₂

310

₁



3

1

 

 







n



n



n



n



n

n

y

y

y

y

y

y

14)



_{5 7}

61

_{5 5}

5

_{3 5}

305

_{3 3}

10

₃

610

₁



6

1

 

 







n



n



n



n



n

n

y

y

y

y

y

y

15)



31

_{5 7}

244

_{5 6}

155

_{3 5}

1220

_{3 4}

310

₃

2440

₂



31

1

 

 







n



n



n



n



n

n

y

y

y

y

y

III. REMARKABLE OBSERVATIONS

A. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbola which are presented in the Table: 2 below:

Table: 2 Hyperbolas

S. No Hyperbola

_

_X

_,

_Y

_

1

₁₅

2 2

₆₀





Y

X



8

x

n1



x

n2

,

31

x

n1



4

x

n2



2

₁₅

2

₁₆

2

₃₈₄₀





Y

X



63

x

n1



x

n3

,

61

x

n1



x

n3



3

₃

2

₅

2

₁₂





Y

X



4

x

n1



5

y

n1

,

3

x

n1



4

y

n1



4 2

₂₄₀

2

₅₇₆





Y

X



93

x

n1



15

y

n2

,

6

x

n1



y

n2



5

₃

2

₅

2

₁₁₅₃₂





Y

X



244

x

n1



5

y

n3

,

189

x

n1



4

y

n3



6

₅

2

₃

2

₁₈₀





Y

X



189

x

n2



24

x

n3

,

244

x

n2



31

x

n3



7

₄₈

2

₅

2

₁₉₂





Y

X



x

n2



10

y

n1

,

3

x

n2



31

y

n1



8

₃

2

₅

2

₁₂





Y

X



31

x

n2



40

y

n2

,

24

x

n2



31

y

n2



9

₄₈

2

₅

2

₁₉₂





Y

X



61

x

n2



10

y

n3

,

189

x

n2



31

y

n3



10

₃

2

₅

2

₁₁₅₃₂





Y

X



4

x

n3



315

y

n1

,

3

x

n3



244

y

n1



11

₃

2

₅

2

₁₉₂





Y

X



31

x

n3



315

y

n2

,

24

x

n3



244

y

n2



12

₃

2

₅

2

₁₂





Y

X



244

x

n3



315

y

n3

,

189

x

n3



244

y

n3



13 2

₁₅

2

₃₆





Y

X



4

y

n2



31

y

n1

,

8

y

n1



y

n2



14

₁₆

2

₁₅

2

₂₃₀₄





Y

X



y

n3



61

y

n1

,

63

y

n1



y

n3



15 2

₁₅

2

₃₆





Y

B. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabola which are presented in the Table: 3 below:

Table: 3 Parabolas

S. No Parabola

_

_X

_,

_Y

_

1

₁₅

2

₃₀





Y

X



8

x

2n2



x

2n3

,

31

x

n1



4

x

n2



2

₁₅

₂

2

₂₄₀





Y

X



63

x

2n2



x

2n4

,

61

x

n1



x

n3



3

₃

₅

2

₆





Y

X



4

x

2n2



5

y

2n2

,

3

x

n1



4

y

n1



4

₂₀

2

₂₄





Y

X



93

x

2n2



15

y

2n3

,

6

x

n1



y

n2



5

₉₃

₅

2

₅₇₆₆





Y

X



244

x

2n2



5

y

2n4

,

189

x

n1



4

y

n3



6

₅

2

₃₀





Y

X



189

x

2n3



24

x

2n4

,

244

x

n2



31

x

n3



7

₄₈

₅

2

₉₆





Y

X



x

2n3



10

y

2n2

,

3

x

n2



31

y

n1



8

₃

₅

2

₆





Y

X



31

x

2n3



40

y

2n3

,

24

x

n2



31

y

n2



9

₄₈

₅

2

₉₆





Y

X



61

x

2n3



10

y

2n4

,

189

x

n2



31

y

n3



10

₉₃

₅

2

₅₇₆₆





Y

X



4

x

2n4



315

y

2n2

,

3

x

n3



244

y

n1



11

₁₂

₅

2

₉₆





Y

X



31

x

2n4



315

y

2n3

,

24

x

n3



244

y

n2



12

₃

₅

2

₆





Y

X



244

x

2n4



315

y

2n4

,

189

x

n3



244

y

n3



13

₅

2

₆





Y

X



4

y

2n3



31

y

2n2

,

8

y

n1



y

n2



14

₃₂

₅

2

₃₈₄





Y

X



y

2n4



61

y

2n2

,

63

y

n1



y

n3



15

₅

2

₆





Y

X



31

y

2n4



244

y

2n3

,

63

y

n2



8

y

n3



IV. CONCLUSION

In this paper, we have presented infinitely many integer solutions for the Diophantine equation, represented by hyperbola is given

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2



(

a



1

)

x

2



3

a



1

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a



2

)

x

2



ay

2



4

a

(

k



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

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k

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,

a

,

k



0

,

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2

X

2



3

Y

2



5

,

American Journal of Applied Mathematics and Mathematical Sciences, 1(2012) 1-4.

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

(

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

1

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y

2



3

a



1

,

Discovery,4(10), 2013,22-24.

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y

2



68

x

2



13

,

International Journal of Advanced Education and Research, 2,2017,59-63.

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y

2



7

x

2



32

,

International Journal of Advanced Science and Research, 2,2017,18-22.

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y

2



8

x

2



16

,

National Journal of Multidisciplinary Research and Development, 2,2017,1-5.

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y

2



12

x

2



13

,

International Journal of Advanced Education and Research, 2, 2017,4-8.

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y

2



102

x

2



33

,

International Journal of Advanced Education and Research, 2(1),2017,91-96.

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y

2



10

x

2



24

,

International Journal of Multidisciplinary Education and Research, 2,2017,34-39.

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(

a



2

)

x

2



ay

2



4

a

(

k



1

)



2

k

2

,

a

,

k



0

,

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x

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

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(

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

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