ON THE NEGATIVE PELL EQUATION 11

(1)

[Mallika, 6(7): July 2019] ISSN 2348 – 8034 DOI- 10.5281/zenodo.3351447 Impact Factor- 5.070

(C)Global Journal Of Engineering Science And Researches 75

G ^LOBAL J ^{OURNAL OF} E NGINEERING S CIENCE AND R ^ESEARCHES

ON THE NEGATIVE PELL EQUATION 11

23

²

2

 x 

y

Dr. S. Mallika

Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India

ABSTRACT

The binary quadratic Diophantine equation represented by the negative pellian

y

²

 23 x

²

 11

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.

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

 23 x

²

 11

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,

11 23

²

2

 x 

y

(1)

The smallest negative integer solutions of (1) are,

0 2

x ,y₀9

The pellian equation is

1 23

²

2

 x 

y

(2) The initial solution of pellian equation is

~ 24 ,

~ 5

0

0 y 

x

The general solution

 x ,

n

y

n



of (2) is given by,

n

g

x 2 23

~  1

,

n

f

y 2

~  1

Where,



²⁴^⁵ ²³

 

^¹^²⁴^⁵ ²³



^¹

 ⁿ ⁿ

fn



²⁴^⁵ ²³

 

^¹^²⁴^⁵ ²³



^¹

 ⁿ ⁿ

gn

n  0 , 1 , 2 ,...

(2)

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 2 23

9

1 



n n

n f g

y 23

23 2

9

1 



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

0

48

₂ ₁

3



_



_



 n n

n

x x

x

0

48

₂ ₁

3



_



_



 n n

n

y y

y

n=0,1,2,3…..

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

n x

n

y

_n

0 2 9

1 93 446

2 4462 21399

3 214083 1026706

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

values are even and odd alternatively.

values are odd and even alternatively.

Each of the following expression is a nasty number:

 110 892

2 2

18

2 4



55 6





 x

_n

x

_n

 880 7133

2 2

3

2 4



440 6





 x

_n

x

_n

 88 713

2 2

3

2 3



44 6





 x

_n

y

_n

 25322 205252

2 2

18

2 4



12661 6





 x

_n

y

_n

 132 23

2 4

223

2 3



11 1





 x

_n

y

_n

 25322 92

2 4

42798

2 3



12661 6





 x

_n

y

_n

 2530 92

2 3

4278

2 2



1265 6





 y

_n

y

_n

(3)

 2640 2

2 4

4462

2 3



220 1





 y

_n

y

_n

 110 42798

2 3

892

2 4



55 6





 x

_n

x

_n

 22 4278

2 3

892

2 3



11 6





 x

_n

y

_n

 132 223

2 4

51313

2 3



11 1





 y

_n

x

_n

 88 713

2 4

7133

2 3



44 6





 x

_n

y

_n

 22 205252

2 4

42798

2 4



11 6





 x

_n

y

_n

 2530 4278

2 4

205252

2 3



1265 6





 y

_n

y

_n

Each of the following expressions is a cubical integer.

 892

3 3

18

3 5

2676

1

54

1



55 1





_n



_n



_n

n

x y x

x

 7133

3 3

3

3 5

21399

1

9

3



440 1





_n



_n



_n

n

x x x

x

 205252

3 3

18

3 5

615756

1

54

3



12661 1





_n



_n



_n

n

y x y

x

 713

3 3

3

3 4

2139

1

9

2



44 1





_n



_n



_n

n

y x y

x

 23

3 5

223

3 4

69

2

669

1



66 1





_n



_n



_n

n

y x y

x

 92

3 5

42798

3 4

276

3

128394

1



12661 1





_n



_n



_n

n

y x y

x

 92

3 4

4278

3 3

276

2

12834

1



1265 1





_n



_n



_n

n

y y y

y



3 5

2231

3 4

3

6693

1



660 1





_n



_n



_n

n

y y y

y

 42798

3 4

892

3 5

128394

2

2676

3



55 1





_n



_n



_n

n

x x x

x

 4278

3 4

892

3 4

12834

2

2676

2



11 1





_n



_n



_n

n

y x y

x

 223

3 5

51313

3 4

669

3

153939

2



66 1





_n



_n



_n

n

x y x

y

(4)

 713

3 5

7133

3 4

2139

3

21399

2



44 1





_n



_n



_n

n

y x y

x

 205252

3 5

42798

3 5

615756

3

128394

3



11 1





_n



_n



_n

n

y x y

x

 4278

3 5

205252

3 4

12834

3

615756

2



1265 1





_n



_n



_n

n

y y y

y

Each of the following expressions is a biquadratic integer.

 ⁸⁹² ¹⁸ ³⁵⁶⁸ ⁷² ³³⁰ 

55 1

4 2 2

2 6

4 4

4_n_

 x

_n_

 x

_n_

 x

_n_



x

 ⁷¹³³ ³ ²⁸⁵³² ¹² ²⁶⁴⁰ 

440 1

4 2 2 2 6

4 4

4_n_

 x

_n_

 x

_n_

 x

_n_



x

 ⁷¹³ ³ ²⁸⁵² ¹² ²⁶⁴ 

44 1

3 2 2

2 5

4 4

4_n_

 y

_n_

 x

_n_

 y

_n_



x

 

 













_ _ _



75966

72 821008

18 205252

12661

1 x

4n 4

y

4n 6

x

2n 2

y

2n 4

 ²³ ²²³ ⁹² ⁸⁹² ³⁹⁶ 

66 1

3 2 4

2 5

4 6

4_n_

 y

_n_

 x

_n_

 y

_n_



x

 ⁹² ⁴²⁷⁹⁸ ³⁶⁸ ¹⁷¹¹⁹² ⁷⁵⁹⁶⁶ 

12661 1

3 2 4

2 5

4 6

4_n_

 y

_n_

 x

_n_

 y

_n_



x

 ⁹² ⁴²⁷⁸ ³⁶ ¹⁷¹¹² ⁷⁵⁹⁰ 

1265 1

2 2 3

2 4

4 5

4_n_

 y

_n_

 y

_n_

 y

_n_



y

 ²²³¹ ⁴ ⁸⁹²⁴ ²⁶⁴⁰ 

660 1

3 2 4

2 5 4 6

4_n_

 y

_n_

 y

_n_

 y

_n_



y

 ⁴²⁷⁹⁸ ⁸⁹² ¹⁷¹¹⁹⁸ ³⁵⁶⁸ ³³⁰ 

55 1

4 2 3

2 6

4 5

4_n_

 x

_n_

 x

_n_

 x

_n_



x

 12426 1242 49704 4968 936 

11 1

4 2 3

2 6

4 5

4_n_

 x

_n_

 y

_n_

 x

_n_



y

 ²²³ ⁵¹³¹³ ⁸⁹² ²⁰⁵²⁵² ³⁹⁶ 

66 1

3 2 4

2 5

4 6

4_n_

 x

_n_

 y

_n_

 x

_n_



y

 ⁷¹³ ⁷¹³³ ²⁸⁵² ²⁸⁵³² ²⁶⁴ 

44 1

3 2 4

2 5

4 6

4_n_

 y

_n_

 x

_n_

 y

_n_



x

 4278 ^y

n

205252 ^y

n

17112 ^y

n

821008 ^y 

1265 1

4 2 5

4 6

4 _



_



_



 

 













_ _ _



66 171192 821008

42798 205252

11 1 x

4n 6

y

4n 6

x

2n 4

y

2n 4

(5)

Each of the following expression is a quintic integer:

 

 

















1 1

5 3 3 3 7

5 5 5

180 8920

90 4460

18 892

55 1

n n

y x

x x

 

 













3 1

5 3 3 3 7

5 5 5

30 71330

15 356655

3 7133

440 1

n n

x x

 

 

















2 1

4 3 3 3 6

5 5 5

30 7130

15 3565

3 713

44 1

n n

y x

 

 

















3 1

5 3 3

3 7

5 5 5

180 2052520

90 1026260

18 205252

12661 1

n n

y x

 

 

















1 2

4 3 5

3 6

5 7

5

2230 230

1115 115

223 23

66 1

n n

y x

 

 

















1 3

4 3 5

3 6

5 7

5

427980 920

213990 460

42798 92

12661 1

n n

y x

 

 

















1 2

3 3 4

3 5

5 6

5

42780 920

21390 460

4278 92

1265 1

n n

y y

 

 

















1 3

4 3 5

3 6 5 7

5

22310 20

11155 5

2231 660

1

n n

y y

 

 

















3 2

5 3 4

3 6

5 6

5

8920 427980

4460 213990

892 42798

55 1

n n

x x

 

 

















2 2

4 3 4

3 7

5 6

5

8920 42780

4460 21390

892 42798

11 1

n n

y x

x x

 

 

















2 3

4 3 5

3 6

5 7

5

513130 2230

256565 1115

51313 223

66 1

n n

x y

 

 

















2 3

4 3 5

3 6

5 7

5

71330 7130

35665 3565

7133 713

44 1

n n

y x

 

 

















3 3

5 3 5

3 7

5 7

5

427980 2052520

213990 1026260

42798 205252

11 1

n n

y x

 

 

















2 3

4 3 5

3 6

5 7

5

2052520 42780

1026260 21390

205252 4278

1265 1

n n

y y

y

(6)

Relations among the solutions are given below:

1 1

2

5

_

24

_





_n



_n

n

y x

x

1 1

3

240

_

1151

_





_n



_n

n

y x

x

1 1

2

24

_

115

_





_n



_n

n

y x

y

1 1

3

1151

_

5520

_





_n



_n

n

y x

y

1 2

3

48

_ _





_n



_n

n

x x

x

1 2

2

24 5 y

_n_

 x

_n_

 x

_n_

1 2

3

1151 24

5 y

_n_

 x

_n_

 x

_n_

1 3

10 y

_n_2

 x

_n_

 x

_n_

1 3

3

1151

240 y

_n_

 x

_n_

 x

_n_

1 2

3

1151 115

24 y

_n_

 y

_n_

 x

_n_

2 1

3

5 1151

24 x

_n_

 y

_n_

 x

_n_

2 1

2

115 24 y

_n_

 y

_n_

 x

_n_

2 1

3

24 5520

24 y

_n_

 y

_n_

 x

_n_

3 1

2

24 115

1151 y

_n_

 y

_n_

 x

_n_

3 1

3

5520

1151 y

_n_

 y

_n_

 x

_n_

1 2

3

5520 115

115 y

_n_

 y

_n_

 y

_n_

2 3

2

24 5 y

_n_

 x

_n_

 x

_n_

2 3

3

24 5 y

_n_

 x

_n_

 x

_n_

2 2

3

24

_

115

_





_n



_n

n

y x

y

3 2

3

115 24 y

_n_

 y

_n_

 x

_n_

Remarkable Observation

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 Y²23X²484

 

 









1 1

18 92

, 18 4

n n

y x

x y

2

Y

²

 23 X

²

 12100

 

 









2 1

1 2

18 892

, 186 4

n n

x x

3 Y²23X² 6969600

 

 









3 1

1 3

9 21399

, 4462 2

n n

x x

x

(7)

4 Y²23X²69696

 

 









2 1

1 2

9 2139

, 446 2

n n

y x

x y

5 Y²23X²641203684

 

 









3 1

1 3

18 205252

, 42798 4

n n

y x

x y

6 Y²23X² 69696

 

 









1 2

2 1

446 46

, 9 93

n n

y x

x y

7

Y

²

 23 X

²

 641203684

 

 









1 3

3 1

42798 92

, 18 8924

n n

y x

x y

8

Y

²

 23 X

²

 3686918400

 

 









2 1

1 2

46 2484

, 518 12

n n

y y

9 Y²23X² 5149497600

 

 









1 3

3 1

102626 46

, 9 21399

n n

y y

10

Y

²

 23 X

²

 12100

 

 









3 2

2 3

892 42798

, 8924 186

n n

x x

11

Y

²

 23 X

²

 484

 

 









2 2

892 4278

, 892 18

n n

y x

x y

12

Y

²

 23 X

²

 69696

 

 









2 3

3 2

102626 446

, 93 21399

n n

x y

y x

13

Y

²

 23 X

²

 69696

 

 









2 3

3 2

21399 2139

, 446 4462

n n

y x

x y

14

Y

²

 23 X

²

 484

 

 









3 3

42798 205252

, 42798 8924

n n

y x

x y

15

Y

²

 23 X

²

 6400900

 

 









2 3

3 2

205252 4278

, 892 42798

n n

y y

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

(8)

Table 3: Parabola

S.NO Parabola (X,Y)

1 11Y23X²484

 

 











22 18

92 , 18 4

2 2 2

2

1 1

n n

y x

x y

2 55Y23X²12100

 

 











110 18

892 , 186 4

3 2 2 2

1 2

n n

x x

3

1320 Y  23 X

²

 6969600

 

 











2640 9

21399

, 4462 2

4 2 2 2

1 3

n n

x x

4 132Y23X²69696

 

 











264 9

2139

, 446 2

3 2 2 2

1 2

n n

y x

x y

5 12661Y23X²641203684

 

 











25322 18

205252

, 42798 4

4 2 2

2

1 3

n n

y x

x y

6 132Y23X² 69696

 

 











264 446

46 , 9 93

2 2 3

2

2 1

n n

y x

x y

7

12661 Y  23 X

²

 641203684

 

 











25322 42798

92 , 18 8924

2 2 4

2

3 1

n n

y x

x y

8

1265 Y  23 X

²

 3686918400

 

 











2530 46

2484

, 518 12

3 2 2

2

1 2

n n

y y

9

30360 Y  23 X

²

 5149497600

 

 











60720 102626

46 , 9 21399

2 2 4

2

3 1

n n

y y

10

55 Y  23 X

²

 12100

 

 











110 892

42798

, 8924 186

4 2 3

2

2 3

n n

x x

11

11 Y  23 X

²

 484

 

 











22 892

4278

, 892 18

3 2 3

2

2 2

n n

y x

x y

12

132 Y  23 X

²

 69696

 

 











264 102626

446 , 93 21399

4 2 4

2

3 2

n n

x y

y

x

ON THE NEGATIVE PELL EQUATION 11

G LOBAL J OURNAL OF E NGINEERING S CIENCE AND R ESEARCHES