ON THE NEGATIVE PELL EQUATION y2=102x2-18

GSJ: Volume 7, Issue 4

,

April

2019, Online: ISSN 2320-9186

www.globalscientificjournal.com

ON

THE

NEGATIVE

PELL

EQUATION

1

D.Maheswari and

2

K.Kaviyarasi

1

Assistant Professor, Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India. email; [email protected]

2

Mphil Scholar, Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India. email;[email protected]

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 and Pythagorean triangle.

KEYWORDS:

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

Introduction

The binary quadratic equations of the form

y

2



Dx

2



1

where D is non-square negative 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



102

x

2



18

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

Method of Analysis

The negative Pell equation representing hyperbola under consideration is,

102

18

2

2





x

y

(1)

The smallest positive integer solutions of (1) are,

x

0



3

,

y

0



30

Consider the pellian equation

102

1

2

2

_

_

x

y

(2)

The initial solution of (2)

10

~

0



x

,

~

y

₀



101

The general solution



x

_n

,

y

_n



of (2) is given by,

n

n

g

x

102

2

1

~

_

_,

n

n

f

y

2

1

~

_

where,

1 1

)

102

10

101

(

)

102

10

101

(













n n

n

f

1 1

)

102

10

101

(

)

102

10

101

(













n n

n

g

Applying Brahmagupta lemma between



x

₀

,

y

₀



and



x

~

_n

,

~

y

_n



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

n n

n

f

g

x

102

2

30

2

3

1







n n

n

f

g

y

102

2

306

2

30

1







The recurrence relation satisfied by the solution and are given by,

x

n3



202

x

n2



x

n1



0

y

n3



202

y

n2



y

n1



0

Some numerical examples of and satisfying (1) are given in the Table 1 below,

Table 1: Examples

N

x

n

y

n

0 3 30

1 603 6090

2 121803 1230150

3 12423906 248484210

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

 values are even.

 Each of the following expression is a nasty number:





18

306

_{2 2}

30

_{2 2}



9

6









x

n

y

n





180

6090

_{2 2}

30

_{2 3}



90

6









x

n

x

n





72720

2460300

₁

60

₃



36360

6









x

n

x

n





3636

123012

_{2 2}

60

_{2 3}



1818

6









x

n

y

n





367218

12423906

_{2 2}

30

_{2 4}



183609

6









x

n

y

n





1818

306

_{2 3}

6090

_{2 2}



909

6









x

n

y

n





367218

306

₃

1230150

₁



183609

6









x

n

y

n





18360

306

_{2 3}

61506

_{2 2}



9180

6









y

n

y

n





3708720

306

_{2 4}

12423906

_{2 2}



1854360

6









y

n

y

n





180

1230150

_{2 3}

6090

_{2 4}



90

6









x

n

x

n





18

61506

_{2 3}

6090

_{2 3}



9

6









x

n

y

n





_y



n n

x

4 2 3

2

6090

12423906

1818

909

6













1818

61506

_{2 4}

1230150

_{2 3}



909

6









x

n

y

n





18360

61506

_{2 4}

12423906

_{2 3}



9180

6









y

n

x

n

 Each of the following expression is a cubical integer.





306

_{3 3}

30

_{3 3}

918

₁

90

₁



9

1

 







n



n



n

n

y

x

y

x





6090

_{3 3}

30

_{3 4}

18270

₁

90

₂



90

1

 







n



n



n

n

x

x

x

x





2460300

_{3 3}

60

_{3 5}

7380900

₁

180

₃



36360

1

 







n



n



n

n

x

x

x

x





123012

_{3 3}

60

_{3 4}

369036

₁

180

₂



1818

1

 







n



n



n

n

y

x

y

x





12423906

_{3 3}

30

_{3 5}

37271718

₁

90

₂



183609

1

 







n



n



n

n

y

x

y

x





306

_{3 4}

6090

_{3 3}

918

₂

18270

₁



909

1

 







n



n



n

n

y

x

y

x





306

_{3 5}

1230150

_{3 3}

918

₃

3690450

₁



183609

1

 







n



n



n

n

y

x

y

x





306

_{3 4}

61506

_{3 3}

918

₂

184518

₁



9180

1

 







n



n



n

n

y

y

y

y





306

_{3 5}

12423906

_{3 3}

918

₃

37271718

₁



1854360

1

 







n



n



n

n

y

y

y

y





1230150

_{3 4}

6090

_{3 5}

3690450

₂

18270

₃



90

1

 







n



n



n

n

x

x

x

x





61506

_{3 4}

6090

_{3 4}

184518

₂

18270

₂



9

1

 







n



n



n

n

y

x

y

x





12423906

_{3 4}

6090

_{3 5}

372711718

₂

18270

₃



909

1

 







n



n



n

n

y

x

y

x





61506

_{3 5\}

1230150

_{3 4}

184518

₃

3690450

₂



909

1

 







n



n



n

n

y

x

y

x





61506

_{3 5\}

12423906

_{3 4}

184518

₃

37271718

₂



9180

1

 







n



n



n

n

y

y

y

y

 Each of the following expression is a biquadratic integer.





306

30

1224

120

54



9

1

2 2 2 2 4 4 4

4n



y

n



x

n



y

n



x





6090

30

24360

120

540



90

1

3 2 2 2 5 4 4

4n



x

n



x

n



x

n



x





2460300

60

9841200

240

218160



36360

1

4 2 2 2 6 4 4

4n



x

n



x

n



x

n



x





123012

60

492048

240

10908



1818

1

3 2 2 2 5 4 4

4n



y

n



x

n



y

n



x





12423906

30

49695624

120

1101654



183609

1

3 1 6 4 4

4n



y

n



x

n



y

n



x





306

6090

1224

24360

5454



909

1

2 2 3 2 4 4 5

4n



y

n



x

n



y

n



x





306

1230150

1224

4920600

1101654



183609

1

2 2 4 2 4 4 6

4n



y

n



x

n



y

n



x





306

61506

1224

246024

55080



9180

1

2 2 3 2 4 4 5

4n



y

n



y

n



y

n



y





306

12423906

1224

49695624

11126160



1854360

1

1 3 2 4 4 6

4n



y

n



y

n



y

n



y





1230150

6090

4920600

24360

540



90

1

4 2 3 2 6 4 5

4n



y

n



x

n



x

n



x





61506

6090

246024

24360

54



9

1

3 2 3 2 6 4 5

4n



x

n



x

n



y

n



x





12423906

6090

49695624

24360

5454



909

1

3 2 6 4 5

4n



y

n



x

n



y

n



x





61506

1230150

246024

4920600

5454



909

1

3 2 4 2 5 4 6

4n



y

n



x

n



y

n



x





61506

12423906

246024

49695624

55080



9180

1

3 2 4 2 5 4 6

4n



y

n



y

n



y

n



y

 Each of the following expression is a quintic integer:





306

_{5 5}

30

_{5 5}

1530

_{3 3}

150

_{3 3}

3060

₁

300

₁



9

1

    





n



n



n



n



n

n

y

x

y

x

y

x





6090

_{5 5}

30

_{5 6}

30450

_{3 3}

150

_{3 4}

60900

₁

300

₂



9

1

    





n



n



n



n



n

n

x

x

x

x

x

x





2460300

_{5 5}

60

_{5 7}

12303500

_{3 3}

300

_{3 5}

24603000

₁

600

₃



36360

1

    





n



n



n



n



n

n

x

x

x

x

x

x





123012

_{5 5}

60

_{5 6}

615060

_{3 3}

300

_{3 4}

1230120

₁

600

₂



1818

1

    





n



n



n



n



n

n

y

x

y

x

y

x





12423906

_{5 5}

30

_{5 7}

62119530

_{3 3}

150

_{3 5}

124239060

₁

300

₂



183609

1

    





n



n



n



n



n

n

y

x

y

x

y

x





306

_{5 6}

6090

_{5 5}

1530

_{3 4}

30450

_{3 3}

3060

₂

60900

₁



909

1

    





n



n



n



n



n

n

y

x

y

x

y

x





306

_{5 7}

1230150

_{5 5}

1530

_{3 5}

7380900

_{3 3}

3060

₃

11071350

₁



183609

1

    





n



n



n



n



n

n

y

x

y

x

y

x





306

_{5 6}

61506

_{5 5}

1530

_{3 4}

307530

_{3 3}

3060

₂

615060

₁



9180

1

    





n



n



n



n



n

n

y

y

y

y

y

y





306

_{5 7}

12423906

_{5 5}

1530

_{3 5}

62119530

_{3 3}

3060

₃

124239060

₁



1854360

1

    





n



n



n



n



n

n

y

y

y

y

y

y





1230150

_{5 6}

6090

_{5 7}

6150750

_{3 4}

30450

_{3 5}

12301500

₂

60900

₃



90

1

    





n



n



n



n



n

n

x

x

x

x

x

x





61506

_{5 6}

6090

_{5 6}

307530

_{3 4}

30450

_{3 4}

615060

₂

60900

₂



9

1

    





n



n



n



n



n

n

y

x

y

x

y

x







3

60900

124239060

30450

62119530

6090

1243906

909

1

2 5 3 4 3 7 5 6

5 



y





x





y





x





y

n

x

_{n n n n n}





61506

_{5 7}

1230150

_{5 6}

307530

_{3 5}

6150750

_{3 4}

615060

₃

12301500

₂



909

1

    





n



n



n



n



n

n

y

x

y

x

y

x



61506

5 7

12423906

5 6

307530

3 5

62119530

3 4

615060

3

124239060

3



9180

1

    





n



n



n



n



n

n

y

y

y

y

y

y

 Relations among the solutions are given below:



x

n2



101

x

n1



10

y

n1



x

_n3



20401

x

_n1



2020

y

_n1



y

_n2



1020

x

_n1



101

y

_n1



y

n_3



206040

x

n_1



20401

n_1



x

_n3



202

x

_n2



x

_n1



10

y

n2



101

x

n2



x

n1



10

y

_n3



20401

x

_n2



101

x

_n1



20

y

_n2



x

_n3



x

_n1



2020

y

_n3



20401

x

_n3



x

_n1



101

y

_n3



1020

x

_n1



20401

y

_n2



101

x

n_3



20401

x

n_2



10

y

n_1



101

y

_n2



1020

x

_n2



y

_n1



y

n3



2040

x

n2



y

n1



20401

y

_n2



1020

x

_n3



101

y

_n1



20401

y

_n3



206040

x

_n3



y

_n1



y

n_3



202

y

n_2



y

n_1



10

y

_n2



x

_n3



101

x

_n2



10

y

n3



101

x

n3



x

n2



y

_n3



1020

x

_n2



101

y

_n2



101

y

_n3



1020

x

_n3



y

_n2

Remarkable Observations

I. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbolas which are presented in table 2 below

Table 2: Hyperbola

S.NO Hyperbola (X,Y)

1

₄

2



₄₁₃₁

2



₁₂₉₆

X

Y



y

n1



10

x

n1

,

306

x

n1



30

y

n1



2 2



₅₁

2



₃₂₄₀₀

X

Y



3

x

n2



603

x

n1

,

6090

x

n1



30

x

n2



3

₄

2



₄₀₈

2



_5288198400

X

Y



3

x

n3



121803

x

n1

,

1230150

x

n1



30

x

n3



4

₄

2



₄₀₈

2



_13220496

X

Y



3

y

n2



6090

x

n1

,

61506

n1



30

y

n2



5 2 2 11

10

348490595

.

1

102







X

Y



3

y

n3



1230150

x

n1

,

12423906

x

n1



30

y

n3



6 2



₁₀₂

2



_3305124

X

Y



603

y

n1



30

x

n2

,

306

x

n2



6090

y

n1



7 2 2 11

10

348490595

.

1

102







X

Y



121803

y

n1



30

x

n3

,

306

x

n3



1230150

y

n1



8 2



₁₀₂

2



_337089600

X

Y



6090

y

n1



30

y

n2

,

306

y

n2



61506

y

n1



9 2 2 13

10

375460404

.

1

102







X

Y



1230150

y

n1



30

y

n3

,

306

y

n3



12423906

y

n1



10 2



₁₀₂

2



₃₂₄₀₀

X

Y



603

x

n3



121803

x

n2

,

1230150

x

n2



6090

x

n3



11 2

_

₁₀₂

2

_

₃₂₄

X

Y



603

y

n2



6090

x

n2

,

61506

x

n2



6090

y

n2



12 2



₁₀₂

2



_3305124

X

Y



603

y

n3



1230150

x

n2

,

12423906

x

n2



6090

y

n3



13 2

_

₁₀₂

2

_

_3305124

X

Y



121803

y

n2



6090

x

n3

,

61506

x

n3



1230150

y

n2



14 2



₁₀₂

2



_337089600

X

Y



1230150

y

n2



6090

y

n3

,

61506

y

n3



12423906

y

n2



15 2

_

₂₃

2

_

_8940100

X

Y



518

y

n3



24852

y

n2

,

119186

y

n2



2484

y

n3



II. Employing linear combination among the solutions of (1), one may generate integer solutions for other choices of parabolas

which are presented in table 3 below

Table 3: Parabola

S.N O

Parabola (X,Y)

1

₄



₄₅₉

2



₁₄₄

X

Y



y

_n1



10

x

_n1

,

306

x

_2n2



30

y

_2n2



18



2

₉₀



₅₁

2



₃₂₄₀₀

X

Y



3

x

n2



603

x

n1

,

6090

x

2n2



30

x

2n3



180



3

₃₆₃₆₀



₄₀₈

2



_5288198400

X

Y



3

x

_n3



121803

x

_n1

,

2460300

x

_2n2



60

x

_2n4



72720



4

₁₈₁₈



₄₀₈

2



_13220496

X

Y



3

y

n2



6090

x

n1

,

123012

x

2n2



60

y

2n3



3636



5



₁₀₂

2



₇₃₄₄₃₆

X

Y



3

y

_n3



1230150

x

_n1

,

12423906

x

_2n2



30

y

_2n4



367218



6

₉₀₉



₁₀₂

2



_3305124

X

Y



603

y

n1



30

x

n2

,

306

x

2n3



6090

y

2n2



1818



7



₁₀₂

2



₇₃₄₄₃₆

X

Y



306

x

_n3



1230150

y

_n1

,

306

x

_n3



1230150

y

_n1



367218



8

₉₁₈₀



₁₀₂

2



_337089600

X

Y



6090

y

n1



30

y

n2

,

306

y

2n3



61506

y

2n2



18360



9 2 13

10

375460404

.

1

102

1854360

Y



X







1230150

y

_n1



30

y

_n3

,

306

y

_2n4



12423906

y

_2n2



3708720



10

₉₀



₁₀₂

2



₃₂₄₀₀

X

Y



603

x

_n3



121803

x

_n2

,

1230150

x

_2n3



6090

x

_2n4



180



11

₉



₁₀₂

2



₃₂₄

X

Y



603

y

_n2



6090

x

_n2

,

61506

x

_2n3



6090

y

_2n3



18



12

₉₀₉



₁₀₂

2



_3305124

X

Y



603

y

_n3



1230150

x

_n2

,

12423906

x

_2n3



6090

y

_2n4



1818



13

₉₀₉

_

₁₀₂

2

_

_3305124

X

Y



121803

y

_n2



6090

x

_n3

,

61506

x

_2n4



1230150

y

_2n3



1818



14

₉₁₈₀



₁₀₂

2



_337089600

X

Y



1230150

y

_n2



6090

y

_n3

,

61506

y

_2n4



12423906

y

_2n3



18360



15

₁₄₉₅



₂₃

2



_8940100

X

Y



518

y

_n3



24852

y

_n2

,

119186

y

_2n3



2484

y

_2n4



2990



SPECIAL CASES:



 

 

 

 

2

1 , 3 2 , 3 2 , 3 6 2

1 , 3 10

18

918







x y



y x

x

y

t

P

t

t

t

P



 

 

 

 

2

1 , 3 2 , 3 2

1 , 3 10 2

, 3 6

18

102

9

P

y

t

x



P

x

t

y



t

x

t

y







   

 





2

2 2 , 3 2 , 3 2 , 3 2 4

1 2

2 2 , 3 10

18

6

102

_{ }





x y



y x

x

y

t

P

t

t

t

P



 





 





2

2 2 , 3 2 , 3 2 2 2 , 3 10 2

, 3 8

1

102

18

36

P

_y

t

_x



P

_x

t

_y



t

_x

t

_y











 





 

2

1 , 3 2 2 2 , 3 2 1 , 3 8

1 2

2 2 , 3 6

18

36

102

9

P

y

t

x



P

x

t

y



t

x

t

y











  











2

2 2 , 3 2 1 , 3 2

2 2 , 3 2 3 2

1 , 3 2 4

1

102

3

18

6

P

y_

t

x_



P

x

t

y_



t

x_

t

y_

Conclusion

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

y

2



102

x

2



18

_{. 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, (1952) New York. 2. L.J.Mordel, Diophantine equations, Academic Press, (1969) New York.

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

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

5. M.A.Gopalan,S.vidhyalakshmi,D.Maheswari,Observations on the hyperpola

y

2



30

x

2



1

IJOER Vol-1,issue 3, (2013),312-314. 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) (2013) 31-34.

7. R.Suganya,D.Maheswari, ‘’On the negative pellian eqution

y

2



110

x

2



29

,Journal of the mathematics and informatics, Vol-11,pp-63-71,2017

8. S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Integral points on the Hyperbola

x

2



4

xy



y

2



15

x



0

, Diophantus J.Maths,1(7) (2014) 338-340.

9. M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, on the integral solutions Of the binary quadratic equation

x

2



15

y

2



11

t, scholars journal of Engineering and technology, 2(2A) (2014) 156-158.

10. M.A.Gopalan, S.Vidhyalakshmi, D.Maheswari, ‚Observations on the hyperbola

y

2



34

x

2



1

,‛ Bulletin of mathematics and statistics research, Vol-2, Issue 4,2014, PP-414-417.