On The Negative Pell Equation Y2 = 30x2 - 45

On The Negative Pell Equation

D. Maheswari

#

1 and A.Mercy Carolin*2

1

#

Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620002, Tamilnadu, India.

2

*

Mphil Scholar, Department of Mathematics,Shrimati Indira Gandhi College, Trichy-620002, Tamilnadu, India.

ABSTRACT:

The binary quadratic Diophantine equation represented by the negative pellian is analyzed for its non-zero distinct solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas.

KEYWORDS: Binary quadratic, Hyperbola, Parabola, Integral solutions, Pell equation.

I. INTRODUCTION

The binary quadratic equations of the form

y

2



Dx

2



1

where D is non-square positive integer has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values[1-4].For an extensive review of various problems, one may refer[5-10]. In this communication, yet another interesting equation given by

y

2



30

x

2



45

is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented.

II. METHOD OF ANALYSIS

The negative Pell equation representing hyperbola under consideration is

45

30

2

2





x

y

(1)

whose smallest positive integer solution is

15

,

3

₀

0



y



x

.

To obtain the other solutions of (1), consider the Pell equation

1

30

2

2





x

y

(2)

whose initial solution is given by

11

~

,

2

~

_

_

n

n

y

x

The general solution



x

~

_n

,

~

y

_n



of (2) is given by,

n n

n

n

g

y

f

x

11

1

~

,

30

2

1

where,



 

1



1

30

2

11

30

2

11













n n

n

f



11



2

30

 

1



11



2

30



1

,





1

,

0

,

1

,

2

...



 

n

g

n n n

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

30

2

1

2

1

1







n n

n

f

g

y

_1





30

The recurrence relations satisfied by x and y are given by

0

22

_{2 1}

3











 n n

n

x

x

x

0

22

_{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 1 2

0 3 15

1 63 345

2 1383 7575

3 30363 166305

4 666603 3651135

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

1.

x

n₁ is always odd,

y

n1is always odd. 2.Relations among the solutions

 22

x

n₂



x

n₁



x

n₃



0



2

y

_n1



x

_n2



11

x

_n1



0



2

y

n₃



241

x

n₂



11

x

n₁



0



x

n₃



241

x

n₃



44

y

n₁



0



2399

y

n₁



14520

x

n₁



y

n₃



0



120

x

_n1



y

_n1



y

_n2



0



60

x

_n1



241

y

_n2



11

y

_n3



0



28980

x

n₁



11

y

n₂



241

y

n₃



0



11

x

n₃



241

x

n₂



2

y

n₁



0



11

x

_n2



x

_n3



2

y

_n2



0



11

x

_n3



x

_n2



2

y

_n3



0



2

y

_n1



x

_n2



11

x

_n1



0



60

x

_n2



11

y

_n2



y

_n1



0



11

y

n₁



1320

x

n₂



11

y

n₃



0



11

y

n₂



60

x

n₂



y

n₃



0



11

x

_n3



241

x

_n3



2

y

_n1



0



11

y

_n1



60

x

_n3



241

y

_n2



0



1320

x

n3



y

n1



241

y

n3



0



2

y

n2



11

x

n2



11

x

n3



0

3. Each of the following expressions represents a nasty number





138

6

12



3

1

3 2 2

2n



x

n



x





3030

6

12



66

1

4 2 2

2n



x

n



x





36

6

12



3

2

2 2 1

2n



y

n



x





11340

90

12



495

2

3 2 2

2n



y

n







16596

6

12



723

2

4 2 2

2n



y

n



x





45450

2070

12



90

2

4 2 3

2n



x

n



x



6



90

345



2

495

2

2 2 3

2n



y

n



x



6



1890

345



2

45

2

4 2 4

2n



y

n



x



6



41490

345



2

495

2

4 2 3

2n



y

n



x



6



90

7575



2

10845

2

2 2 4

2n



y

n



x



6



1890

7575



2

495

2

3 2 4

2n



y

n



x





41490

7575



2

45

2

4 2 4

2n



y

n



x



6



1890

90



2

2700

2

3 2 2

2n



y

n



y



6



41490

90



2

59400

2

4 2 2

2n



y

n



y



6



41490

1890



2

2700

2

4 2 3

2n



y

n



y

4. Each of the following expressions represents a cubical integer





23

_{3 3 3 4}

69

₁

3

₂



3

1

 







n



n



n

n

x

x

x

x





505

_{3 3 3 5}

1515

₁

3

₃



66

1

  





n



n



n

n

x

x

x

x





6

_{3 3 3 3}

18

₁

3

₁



3

2

 







n



n



n

n

y

x

y

x





1890

_{3 3}

15

_{3 4}

5670

₁

45

₂



495

2

 







n



n



n

n

y

x

y

x





2766

_{3 3 3 5}

8298

₁

3

₃



723

2

 







n



n



n

n

y

x

y





7575

_{3 4}

345

_{3 5}

22725

₁

1035

₃



90

2

  





n



n



n

n

x

x

x

x





90

_{3 4}

345

_{3 3}

270

₂

1035

₁



495

2

 







n



n



n

n

y

x

y

x





1890

_{3 4}

345

_{3 4}

5670

₂

1035

₂



45

2

 







n



n



n

n

y

x

y

x





41490

_{3 4}

345

_{3 5}

124470

₂

1035

₃



495

2

 







n



n



n

n

y

x

y

x





90

_{3 5}

7575

_{3 3}

270

₃

22725

₁



10845

2

 







n



n



n

n

y

x

y

x





1890

_{3 5}

7575

_{3 4}

5670

₃

227253

₂



495

2

 







n



n



n

n

y

x

y

x





41490

_{3 5}

7575

_{3 5}

124470

₃

22725

₃



45

2

 







n



n



n

n

y

x

y

x





1890

_{3 3}

90

_{3 4}

5670

₁

270

₂



2700

2

 







n



n



n

n

y

y

y

y





41490

_{3 3}

90

_{3 5}

124470

₁

270

₃



59400

2

 







n



n



n

n

y

y

y

y





41490

_{3 4}

1890

_{3 5}

124470

₂

5670

₃



2700

2

 







n



n



n

n

y

y

y

y

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





23

92

4

18



3

1

3 2 2 2 5 4 4

4n



x

n



x

n



x

n



x





505

2020

4

396



66

1

4 2 2 2 6 4 4

4n



x

n



x

n



x

n



x





6

24

4

18



3

2

2 2 2 2 4 4 4

4n



y

n



x

n



y

n



x





1890

15

7560

60

2970



495

2

3 2 2 2 5 4 4

4n



y

n



x

n



y

n



x





2766

1064

4

4338



723

2

4 2 2 2 6 4 4

4n



y

n



x

n



y

n



x





7575

345

30300

1380

540



90

2

4 2 3 2 6 4 5





20

345

360

1380

2970



495

2

2 2 3 2 4 4 5

4n



y

n



x

n



y

n



x





1890

345

7560

1380

2970



45

2

3 3 5 4 5

4n



y

n



x

n



y

n



x





41490

345

165960

1380

2970



495

2

4 2 3 2 6 4 5

4n



y

n



x

n



y

n



x





90

7575

360

30300

65070



10845

2

2 2 4 2 4 4 6

4n



y

n



x

n



y

n



x





1890

7575

7560

30300

2970



495

2

3 2 4 2 5 4 6

4n



y

n



x

n



y

n



x





41490

7575

165960

30300

2970



45

2

4 2 4 2 6 4 6

4n



y

n



x

n



y

n



x





1890

90

7560

360

16200



2700

2

3 2 3 2 5 4 4

4n



y

n



y

n



y

n



y





41490

90

165960

360

356400



59400

2

4 2 2 2 6 4 4

4n



y

n



y

n



y

n



y





41490

1890

165960

7560

16200



2700

2

4 2 3 2 6 4 5

4n



y

n



y

n



y

n



y

6. Each of the following expressions represents a quintic integer





23

_{5 5 5 6}

115

_{3 3}

5

_{3 4}

460

₁

20

₂



3

1

    





n



n



n



n



n

n

x

x

x

x

x

x





505

_{5 5 5 7}

2525

_{3 4}

5

_{3 5}

10100

₁

20

₃



66

1

    





n



n



n



n



n

n

x

x

x

x

x

x





6

_{5 5 5 7}

30

_{3 3}

5

_{3 5}

120

₁

20

₁



3

2

    





n



n



n



n



n

n

y

x

y

x

y

x





1890

_{5 5}

15

_{5 6}

9450

_{3 3}

75

_{3 4}

37800

₁

300

₁



495

2

    





n



n



n



n



n

n

y

x

y

x

y

x





2766

_{5 5 5 7}

13830

_{3 3}

5

_{5 5}

55320

₁

20

₂



723

2

    





n



n



n



n



n

n

y

x

y

x

y

x





7575

_{5 6}

345

_{5 7}

37875

_{3 4}

1725

_{3 5}

151500

₁

6900

₃



90

2

    





n



n



n



n



n

n

y

x

x

x

y

x





90

_{5 6}

345

_{5 5}

490

_{3 4}

1725

_{3 5}

1800

₂

6900

₂



495

2

    





n



n



n



n



n

n

y

x

y

x

y





1890

_{5 6}

345

_{5 6}

9450

_{3 4}

1725

_{3 4}

37800

₂

3450

₂



45

2

 

 







n



n



n



n



n

n

y

x

y

x

y

x





41490

_{5 6}

345

_{5 7}

207450

_{3 4}

1725

_{3 5}

829800

₂

6900

₃



495

2

 

 







n



n



n



n



n

n

y

x

y

x

y

x





90

_{5 7}

7575

_{5 5}

40

_{3 5}

37875

_{3 3}

1800

₃

151500

₁



10845

2

 

 







n



n



n



n



n

n

y

x

y

x

y

x





1890

_{5 7}

7575

_{5 7}

9450

_{3 5}

37875

_{3 4}

37800

₃

1175490

₂



495

2

 

 







n



n



n



n



n

n

y

x

y

x

y

x





41490

_{5 7}

7575

_{5 7}

207450

_{3 5}

37875

_{3 5}

829800

₃

151500

₃



45

2

 

 







n



n



n



n



n

n

y

x

y

x

y

x





1890

_{5 5}

90

_{5 6}

9450

_{3 3}

450

_{3 4}

37800

₁

1800

₂



2700

2

 

 







n



n



n



n



n

n

y

y

y

y

y

y





41490

_{5 3}

90

_{5 7}

450

_{3 5}

207450

_{3 3}

829800

₁

1800

₃



59400

2

 

 







n



n



n



n



n

n

y

y

y

y

y

y





41490

_{5 6}

1890

_{5 7}

207450

_{3 5}

9450

_{3 4}

829800

₂

37800

₃



2700

2

 

 







n



n



n



n



n

n

y

y

y

y

y

y

REMARKABLE OBSERVATIONS:

1. 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

₆

2



₅

2



₁₈₀

Y

X



23

x

n₁



x

n₂

,

x

n₂



21

x

n₁



2

₆

2

_

₅

2

_

₈₇₁₂₀

Y

X



505

x

n₁



x

n₃

,



461

x

n₁



x

n₃



3

₆

2

_

₅

2

_

₄₅

Y

X



6

x

n₁



y

n₁

,



5

x

n₁



y

n₁



4

₆₀

2

_

₅

2

_

₄₉₀₀₅₀

Y

X



1890

x

n₁



15

y

n₂

,

3

y

n₂



345

x

n₁



5

₂₄

2

_

₂₀

2

_

_10454580

Y

6

₆₀

2



₂

2



₁₆₂₀₀

Y

X



7575

x

n2



345

x

n3

,



1383

x

n2



63

x

n3



7

₆₀

2



₂

2



₂₄₅₀₂₅

Y

X



90

x

n2



345

y

n1

,

63

y

n1



15

x

n2



8

₆₀

2



₂

2



₄₀₅₀

Y

X



1890

x

n2



345

y

n1

,

63

y

n2



345

x

n2



9

₆₀

2



₂

2



₄₉₀₀₅₀

Y

X



41490

x

n2



345

y

n3

,

63

y

n3



7575

x

n2



10

₆₀

2



₂

2



_235228050

Y

X



90

x

n3



7575

y

n1

,

383

y

n1



15

x

n3



11

₆₀

2



₂

2



₄₉₀₀₅₀

Y

X



1890

x

n3



7575

y

n2

,

1383

y

n2



345

x

n3



12

₆₀

2



₂

2



₄₀₅₀

Y

X



41490

x

n3



7575

y

n3

,

1383

y

n3



7575

x

n3



13

₆₀

2



₂

2



_14580000

Y

X





90

y

n2



1890

y

n1

,



345

y

n1



15

y

n2



14

₆₀

2

_

₂

2

_

_7056720000

Y

X



90

y

n₃



41490

y

n₁

,



7575

y

n₁



15

y

n₃



15

₆₀

2

_

₂

2

_

_14580000

Y

X



1890

y

n₃



41490

y

n₂

,

7575

y

n₂



345

y

n₃



2. 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

₂



₅

2



₃₀

X

Y



23

x

2n2



x

2n3

,

x

n2



21

x

n1



2

₁₁₀



₂

2



₁₄₅₂₀

X

Y



505

x

2n1



x

2n3

,

461

x

n3



x

n3



3

₁₀



₈

2



₃₀

X

Y



6

x

2n1



y

2n1

,

5

x

n1



y

n1



4

₆₆



₈

2



₃₂₆₇₀

X

Y



1890

x

2n1



15

y

2n2

,

3

y

n2



345

x

n1



5

₃₆₁₅



₁₂

2



_2613645

X

Y



2766

x

2n1



y

2n3

,

y

n3



2525

x

n1



6

₉₀



₆₀

2



₈₁₀₀

X

Y



7575

x

2n2



345

x

2n3

,

1383

x

n2



63

x

n3



7

₉₉₀



₁₂₀

2



₄₉₀₀₅₀

X

Y



90

x

2n2



345

y

2n1

,

63

y

n1



15

x

n2



8

₄₅



₆₀

2



₂₀₂₅

X

9

₁₉₈



₂₄

2



₉₈₀₁₀

X

Y



41490

x

2n1



345

y

2n3

,

63

y

n3



7575

x

n2



10

₁₀₈₄₅



₆₀

2



_117614025

X

Y



90

x

2n3



7575

y

2n2



,

1383

y

n1



15

x

n3



11

₁₉₈

_

₂₄

2

_

₉₈₀₁₀

X

Y



1890

x

₂n₃



7575

y

₂n₂

,

1383

y

n₂



345

x

n₃



12

₁₈

_

₂₄

2

_

₈₁₀

X

Y



41490

x

2n3



7575

y

2n3

,

1383

y

n3



7575

x

n3



13

₁₀₈₀



₂₄

2



_2916000

X

Y



25

y

2n4



949

y

2n3



108

,

3001

y

n2



79

y

n3



14

₂₃₇₆₀



₂₄

2



_141344000

X

Y



41490

y

₂n₁



90

y

₂n₃

,

7575

y

n₁



15

y

n₃



15

₁₀₈₀



₂₄

2



_2916000

X

Y



41490

y

2n2



1890

y

2n3

,

7575

y

n2



345

y

n3



III. CONCLUSION

In this paper, we have presented infinitely many integer solutions for the negative Pell Equation

45

30

2

2





x

y

.As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of Pell Equations and determine their integer solutions along with suitable properties.

REFERENCES

[1] L.E.Dickson, History of theory of number, Chelsa publishing company, vol-2, New York, 1952.

[2] L.J.Mordel, Diophantine equations, Academic Press, New York, 1969.

[3] S.J.Telang, Number Theory, Tata McGraw Hill Publishing company Limited, New Delhi,2000.

[4] D.M.Burton, Elementary Number Theory, Tata McGraw Hill Publishing company Limited, New Delhi, 2002.

[5] M.A.Gopalan,S.Vidhyalakshmi and D.Maheswari, “Observations on the hyperbola ,IJOER, Vol-1, Issue

3,312-314, 2013,.

[6] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, Observations on the Hyperbola

10

y

2



3

x

2



13

, Archimedes

J.Math.,3(1), 31-34, 2013.

[7] R.Suganya,D.Maheswari, “ On the negative pellian equation ,Journal of mathematics and informatics,

Vol-11,63-71,2017.

[8] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Observations on the Hyperbola

ax

2





a



1



y

2



3

a



1

, Discovery,

4(10), 22-24, 2013.

[9] M.A.Gopalan, S.Vidhyalakshmi and D.Maheswari, Observations on the Hyperbola ,Bulletin of mathematics and

statistics research, Vol-2, Issue 4, 414-417,2014.

[10] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Integral points on the Hyperbola

x

2



4

xy



y

2



15

x



0

,