ON THE NEGATIVE PELL EQUATION

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On The Negative Pell Equation

y

2



5

x

2



4

K.Meena1, S.Vidhyalakshmi2 and G.Dhanalakshmi3

1

Former VC, Bharathidasan University, Trichy, Tamilnadu, India. 2

Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamilnadu, India. 3

M.Phil scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamilnadu, India.

Article Received: 24 July 2017 Article Accepted: 08 August 2017 Article Published: 11 August 2017

1.INTRODUCTION

Diophantine equation of the form

1 2

2 _{ }

Dx

y , where D is a given positive square-free integer is known as pell equation and is one of the oldest Diophantine equation that has interesting mathematicians all over the world, since antiquity, J.L. Lagrange proved that the positive pell equation y2  Dx2 1 has infinitely many distinct integer solutions whereas the negative pell equation

1 2

2  

Dx

y does not always have a solution. In [1], an elementary proof of a ceriterium for the solvability of the pell equation x2 Dy2 1

where D is any positive non-square integer has been presented. For examples the equations

, 1 3 2

2 _{ }

x

y y2 7x2 4 have no integer solution whereas y2 54x2 1 ,

1 202 2

2 _{ }

x

y have integer solutions. In this context, one may refer [2-17]. More specifically, one may refer “The on-line encyclopedia of integer sequences” (A031396, A130226, A031398) for values of D for which the negative pell equation y2 Dx2 1 is solvable or not. In this communication, the negative pell equation given by is considered and infinitely many integer solutions are obtained. A few interesting relations among the solutions are presented.

2. METHOD OF ANALYSIS

The negative pell equation representing hyperbola under consideration is,

5 4

2

2  

x

y (1)

Whose smallest positive integer solution is x₀ 1,y₀ 1.

To obtain the other solutions of (1), consider the pell equation y2 5x2 1 whose solution is given by

n n g x 5 2 1 ~ _{ ,} n n f y 2 1 ~ _ Where



 



1 1 5 4 9 5 4

9    

 n n

n f



 



1 1 5 4 9 5 4

9    

 n n

n g

Applying Brahamagupta Lemma between



x₀,y₀



and



xn yn



~ ,

~ _{, the other integer solutions of (1) are given by}

2 5xn1  5fn gn, n n

n f g

y 5

2 _1  

Where n= 0,1,2,…..

The recurrence relations satisfied by the solutions x

and y are given by

18 0.

, 0 18 3 2 1 3 2 1             n n n n n n y y y x x x

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

A B S T R A C T

The binary quadratic equation represented by the negative pelliany2 5x2 4 is analyzed for its distinct integer 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 special Pythagorean triangle.

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Table I: Examples

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

1. Both the values of xnand yn are odd.

2. Each of the following expressions is a nasty number.



4

48 3

87x_2n2  x_2n3 



24

288 521x_2n2 x_2n4 



3

36 65x_2n2 y_2n3 



161

1932 3

3495x_2n2  y_2n4 



4

48 87

1563x_2n3 x_2n4 



6

72 58

10x_2n3  y_2n2 

 195x_2n3 87y_2n3 12



3

36 29

1165x_2n3  y_2n4 



161

1932 1563

15x_2n4  y_2n2 



3

36 521

65x_2n4  y_2n3 

 3495x_2n41563y_2n412



24

288 233 _{2 2} 4

2n  y n 

y



4

48 699

39y_2n4  y_2n3

3 .Each of the following expressions is a cubical integer.



8

3 87

29x_3n3 x_3n4  x_n1 x_n2



144

3 1563

521x3n3x3n5  xn1  xn3



18

3 195

65x3n3 y3n4  xn1 yn2



322

3 3495

1165x3n3 y3n5  xn1 yn3



8

87 1563

29

521x_3n4  x_3n5  x_n2  x_n3



18

87 15

29

5x_3n4  y_3n3  x_n2  y_n1



2

87 195

29

65x_3n4  y_3n4  x_n2  y_n2



18

87 3495

29

1165x_3n4  y_3n5  x_n2  y_n3



322

1563 15

521

5x_3n5  y_3n3  x_n3  y_n1



18

1563 195

521

65x_3n5  y_3n4  x_n3  y_n2



2

1563 3495

521

1165x3n5  y3n5  xn3  yn3



8

39 3

13 3 3 2 1

4

3n  y n  yn  yn

y



144

699 3

233 3 3 3 1

5

3n  y n  yn  yn

y

4. Relations among the solutions.

 xn_3 18xn_2xn_1

 4yn_1 xn_2 9xn_1

 4yn_2 9xn_2 xn_1

 4yn_3 161xn_2 9xn_1

 72yn_1 xn_3161xn_1

 72yn_2 9xn_39xn_1

 xn_3 8yn_2 xn_1

9yn_1  yn_220xn_1

161xn_3 72yn_3xn_1

 yn_1 9yn_220xn_2

4yn_3 72yn_24yn_1

 yn_3 20xn_2 9yn_2

n xn yn

0 1 1

1 13 29

2 233 521

3 4181 9349

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161y_n1  y_n3360x_n1

1 3

2 9 20

161y_n  y_n  x_n

9x_n3 161x_n24y_n1

9y_n3 360x_n2y_n1

 x_n3 9x_n24y_n2

Table II: Hyperbolas

9xn_3  xn_24yn_3

161yn_3 360xn_3yn_1

9yn_3 20xn_3yn_2

3. 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 the Table II below.

S. No (X,Y) Hyperbola

1



5x_n2 13 5x_n1,29x_n1 x_n2



Y2 X2 256

2



5x_n3 233 5x_n1,521x_n1 x_n3



Y2 X2 82944

3



5yn2 29 5xn1,65xn1 yn2



1296

2

2  

X Y

4



5yn3 521 5xn1,1165xn1 yn3



414736 2

2_{ }

X Y

5



13 5xn3 233 5xn2,521xn2 29xn3



259 2

2  

X Y

6



13 5yn1 5xn2,5xn2 29yn1



1296

2

2  

X Y

7



13 5yn2 29 5xn1,65xn2 29yn2



16 2

2 _{ }

X Y

8



13 5yn3 521 5xn2,1165xn2 29yn3



1296 2

2  

X Y

9



233 5y_n1 5x_n3,5x_n3 521y_n1



Y2X2 414736

10



233 5yn2 29 5xn3,65xn3 521yn2



1296 2

2  

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ii) Employing linear combinations among the

solutions for other choices of parabolas which are

Table III: Parabolas

solutions of (1), one may generate integer

presented in the Table III below.

11



233 5yn3 521 5xn3,1165xn3 521yn3



16 2

2 

X Y

12



29yn1yn2,yn2 13yn1



5 1280

2

2_{ }

X Y

13



521yn1yn3,yn3233yn1



5 414720 2

2_{ }

X Y

14



521yn229yn3,13yn32333yn2



5Y2X2 1280

S. No (X,Y) Parabola

1



5x_n2 13 5x_n1,29x_2n2 x_2n3



X2 8Y 128

2



5xn3 233 5xn1,521x2n2 x2n4



144 41472

2  

Y X

3



5yn2 29 5xn1,65x2n2 y2n3



18 648

2  

Y X

4



5yn3 521 5xn1,1165x2n2 y2n4



322 207368

2  

Y X

5



13 5xn3 233 5xn2,521x2n3 29x2n4



8 256

2  

Y X

6



13 5yn1  5xn2,5x2n3 29y2n2



18 648

2  

Y X

7



13 5y_n2 29 5x_n1,65x_2n3 29y_2n3



2 2 8

Y X

8



13 5yn3 521 5xn2,1165x2n3 29y2n4



18 648

2  

Y X

9



233 5yn1  5xn3,5x2n4 521y2n2



322 207368

2 _{ }

Y X

10



233 5yn2 29 5xn3,65x2n4 521y2n3



18 648

2  

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iii) Consider mx_n1 y_n1,n x_n1, observe that

. 0  n

m Treat m,n as the generators of the

Pythagorean triangle T



,,



,

mn 2 

 , m2 n2, m2 n2.

Then the following interesting relations are

observed.

a) 23 5 8

b)

P A

20 8 2 7   

c) 2 x_n1y_n1 P

A

4. CONCLUSION

In this paper, we have presented infinitely many

integer solutions for all hyperbola represented

by the negative pell equationy2 5x2 4. As the

binary quadratic Diophantine equations are

rich in variety, one may search for the other choices

of negative pell equations and determine their

integer solutions along with suitable properties.

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

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

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

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