• No results found

On the Positive Pell Equation

N/A
N/A
Protected

Academic year: 2020

Share "On the Positive Pell Equation"

Copied!
8
0
0

Loading.... (view fulltext now)

Full text

(1)

On the Positive Pell Equation

Dr. A. Kavitha1, K. Janani2

1Assistant Professor, 2M. Phil Scholar, Department of Mathematics, Shrimati indira Gandhi college, Trichy-620002, Tamilnadu, India.

Abstract: The binary quadratic Diophantine equation

y

2

17

x

2

8

is analyzed for its non-zero distinct integral solutions. A few interesting relations among the solutions are given. Further, employing the solutions have obtained solutions of other choices of hyperbolas and 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

17

x

2

8

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

II. METHODOFANALYSIS

The positive Pell equation representing hyperbola under consideration is,

8

17

2 2

x

y

(1)

The smallest positive integer solutions of (1) are,

x

0

1

,

y

0

5

D

17

consider the pellian equation is

17

1

2 2

x

y

(2) The initial solution of pellian equation is

8

~

0

x

,

~

y

0

33

,

The general solution

x

~

n

,

~

y

n

of (2) is given by,

n

n

g

x

17

2

1

~

,

n

n

f

y

2

1

~

Where,

1 1

)

17

8

33

(

)

17

8

33

(

n n

n

f

1 1

)

17

8

33

(

)

17

8

33

(

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

17

2

5

2

1

1

n n

n

f

g

y

17

2

17

2

5

1

The recurrence relation satisfied by the solution

x

and

y

are given by,

(2)

Where n=0,1,2,3...

[image:2.612.191.419.109.221.2]

Some numerical examples of

x

n and

y

n satisfying (1) are given in the Table 1 below,

Table 1: Examples n

x

n

y

n

0 1 5 1 73 301 2 4817 19861 3 317849 1310525 4 20973217 86474789

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

A. Both

x

n and

y

n values are odd .

B. Each Of The Following Expression Is A Nasty Number

1)

8

5

2 2

17

2 2

2

3

y

n

x

n

2)

64

5

2 3

301

2 2

16

3

x

n

x

n

3)

4224

5

2 4

19861

2 2

352

1

x

n

x

n

4)

264

5

2 3

1241

2 2

22

1

y

n

x

n

5)

17416

5

2 4

81889

2 2

4354

3

y

n

x

n

6)

264

301

2 2

17

2 3

22

1

y

n

x

n

7)

17416

19861

2 2

17

2 4

4354

3

y

n

x

n

8)

1088

1241

2 2

17

2 3

272

3

y

n

y

n

9)

71808

81889

2 2

17

2 4

5984

1

y

n

y

n

10)

8

301

2 3

1241

2 3

2

3

y

n

x

n

11)

64

301

2 4

19861

2 3

16

3

x

n

x

n

12)

264

301

2 4

81889

2 3

22

1

y

n

x

n

13)

264

19861

2 3

1241

2 4

22

1

y

n

x

n

14)

8

19861

2 4

81889

2 4

2

3

(3)

15)

1088

81889

2 3

1241

2 4

272

3

y

n

y

n

C. Each Of The Following Expressions Is A Cubical Integer

1)

5

3 3

17

3 3

15

1

51

1

4

1

 

n

n

n

n

x

y

x

y

2)

5

3 4

301

3 3

15

2

903

1

32

1

 

n

n

n

n

x

x

x

x

3)

5

3 5

19861

3 3

15

3

59583

1

2112

1

 

n

n

n

n

x

x

x

x

4)

5

3 4

1241

3 3

15

2

3723

1

132

1

 

n

n

n

n

x

y

x

y

5)

5

3 5

81889

3 3

15

3

245667

1

8708

1

 

n

n

n

n

x

y

x

y

6)

301

3 3

17

3 4

903

1

51

2

132

1

 

n

n

n

n

x

y

x

y

7)

19861

3 3

17

3 5

59583

1

51

3

8708

1

 

n

n

n

n

x

y

x

y

8)

1241

3 3

17

3 4

3723

1

51

2

544

1

 

n

n

n

n

y

y

y

y

9)

81889

3 3

17

3 5

245667

1

51

3

35904

1

 

n

n

n

n

y

y

y

y

10)

301

3 5

19861

3 4

903

3

59583

2

32

1

 

n

n

n

n

x

x

x

x

11)

301

3 4

1241

3 4

903

2

3723

2

4

1

 

n

n

n

n

x

y

x

y

12)

301

3 5

81889

3 4

903

3

245667

2

132

1

 

n

n

n

n

x

y

x

y

13)

19861

3 4\

1241

3 5

59583

2

3723

3

132

1

 

n

n

n

n

x

y

x

y

14)

19861

3 5\

81889

3 5

59583

3

245667

3

4

1

 

n

n

n

n

x

y

x

y

15)

81889

3 4\

1241

3 5

245667

2

3723

3

544

1

 

n

n

n

n

y

y

y

y

D. Each Of The Following Expressions Is A Biquadratic Integer

1)

5

17

20

68

24

4

1

2 2 2 2 4 4 4

4n

x

n

y

n

x

n

y

2)

5

301

20

1204

192

32

1

2 2 3 2 4 4 5

4n

x

n

x

n

x

n

(4)

3)

5

19861

20

79444

12672

2112

1

2 2 4 2 4 4 6

4n

x

n

x

n

x

n

x

4)

5

1241

20

4964

792

132

1

2 2 3 2 4 4 5

4n

x

n

y

n

x

n

y

5)

5

81889

20

327556

52248

8708

1

2 2 4 2 4 4 6

4n

x

n

y

n

x

n

y

6)

301

17

1204

68

792

132

1

3 2 2 2 5 4 4

4n

x

n

y

n

x

n

y

7)

19861

17

79444

68

52248

8708

1

4 2 2 2 6 4 4

4n

x

n

y

n

x

n

y

8)

1241

17

4964

68

3264

544

1

3 2 2 2 5 4 4

4n

y

n

y

n

y

n

y

9)

81889

17

327556

68

215424

35904

1

4 2 2 2 6 4 4

4n

y

n

y

n

y

n

y

10)

301

19861

1204

79444

192

32

1

3 2 4 2 5 4 6

4n

x

n

x

n

x

n

x

11)

301

1241

1204

4964

24

4

1

3 2 3 2 5 4 5

4n

x

n

y

n

x

n

y

12)

301

81889

1204

327556

792

132

1

3 2 4 2 5 4 6

4n

x

n

y

n

x

n

y

13)

19861

1241

79444

4964

792

132

1

4 2 3 2 6 4 5

4n

x

n

y

n

x

n

y

14)

19861

81889

79444

327556

24

4

1

4 2 4 2 6 4 6

4n

x

n

y

n

x

n

y

15)

81889

1241

327556

4964

3264

544

1

4 2 3 2 6 4 5

4n

y

n

y

n

y

n

y

E. Each Of The Following Expression Is A Quintic Integer

1)

5

5 5

17

5 5

25

3 3

85

3 3

50

1

170

1

4

1

    

n

n

n

n

n

n

x

y

x

y

x

y

2)

5

5 6

301

5 5

25

3 4

1505

3 3

50

2

3010

1

32

1

    

n

n

n

n

n

n

x

x

x

x

x

x

3)

5

5 7

19861

5 5

25

3 5

99305

3 3

50

3

198610

1

2112

1

    

n

n

n

n

n

n

x

x

x

x

x

x

4)

5

5 6

1241

5 5

25

3 4

6205

3 3

50

2

12410

1

132

1

    

n

n

n

n

n

n

x

y

x

y

x

y

5)

5

5 7

81889

5 5

25

3 5

409445

3 3

50

3

818890

1

8708

1

    

n

n

n

n

n

n

x

y

x

y

x

y

6)

301

5 5

17

5 6

1505

3 3

85

3 4

3010

1

170

2

132

1

    

n

n

n

n

n

n

x

y

x

y

x

(5)

7)

1241

5 5

17

5 6

6205

3 3

85

3 4

12410

1

170

2

544

1

 

 

n

n

n

n

n

n

y

y

y

y

y

y

8)

4817

5 5 5 7

24085

3 3

5

3 5

48170

1

10

3

2112

1

 

 

n

n

n

n

n

n

y

y

y

y

y

y

9)

301

5 7

19861

5 6

1505

3 5

99305

3 4

3010

3

198610

2

32

1

 

 

n

n

n

n

n

n

x

x

x

x

x

x

10)

301

5 6

1241

5 6

1505

3 4

6205

3 4

3010

2

12410

2

4

1

 

 

n

n

n

n

n

n

x

y

x

y

x

y

11)

301

5 7

81889

5 6

1505

3 5

409445

3 4

3010

3

818890

2

132

1

 

 

n

n

n

n

n

n

x

y

x

y

x

y

12)

19861

5 6

1241

5 7

99305

3 4

6205

3 5

198610

2

12410

3

132

1

 

 

n

n

n

n

n

n

x

y

x

y

x

y

13)

19861

5 7

81889

5 7

99305

3 5

409445

3 5

198610

3

818890

3

4

1

 

 

n

n

n

n

n

n

x

y

x

y

x

y

14)

81889

5 6

1241

5 7

409445

3 4

6205

3 5

818890

2

12410

3

544

1

 

 

n

n

n

n

n

n

y

y

y

y

y

y

F. Relations Among The Solutions Are Given Below 1)

x

n2

8

y

n1

33

x

n1

2)

x

n3

528

y

n1

2177

x

n1

3)

y

n2

33

y

n1

136

x

n1

4)

y

n3

2177

y

n1

8976

x

n1

5)

x

n3

66

x

n2

x

n1

6)

8

y

n2

33

x

n2

x

n1

7)

8

y

n3

2177

x

n2

33

x

n1

8)

16

y

n2

x

n3

x

n1

9)

528

y

n3

2177

x

n3

x

n1

10)

33

y

n3

2177

y

n2

136

x

n1

11)

33

x

n3

8

y

n1

2177

x

n2

12)

33

y

n2

y

n1

136

x

n2

13)

y

n3

y

n1

272

x

n2

14)

2177

y

n2

33

y

n1

136

x

n3

15)

2177

y

n3

y

n1

8976

x

n3

16)

y

n3

66

y

n2

y

n1

17)

8

y

n2

x

n3

33

x

n2

(6)

19)

y

n3

33

y

n2

136

x

n2

20)

33

y

n3

y

n2

136

x

n3

III. REMARKABLE OBSERVATIONS

[image:6.612.61.554.180.722.2]

A. 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 2

17

2

64

X

Y

5

x

n1

y

n1

,

5

y

n1

17

x

n1

2 2

17

2

4096

X

Y

73

x

n1

x

n2

,

5

x

n2

301

x

n1

3 2

17

2

17842176

X

Y

4817

x

n1

x

n3

,

5

x

n3

19861

x

n1

4 2

17

2

69696

X

Y

301

x

n1

y

n2

,

5

y

n2

1241

x

n1

5 2

17

2

303317056

X

Y

19861

x

n1

y

n3

,

5

y

n3

81889

x

n1

6 2

17

2

69696

X

Y

5

x

n2

73

y

n1

,

301

y

n1

17

x

n2

7 2

17

2

303317056

X

Y

5

x

n3

4817

y

n1

,

19861

y

n1

17

x

n3

8 2

17

2

1183744

X

Y

5

y

n2

301

y

n1

,

1241

y

n1

17

y

n2

9 2

17

2

5156388864

X

Y

5

y

n3

19861

y

n1

,

81889

y

n1

17

y

n3

10 2

17

2

4096

X

Y

4817

x

n2

73

x

n3

,

301

x

n3

19861

x

n2

11 2

17

2

64

X

Y

301

x

n2

73

y

n2

,

301

y

n2

1241

x

n2

12 2

17

2

69696

X

Y

19861

x

n2

73

y

n3

,

301

y

n3

81889

x

n2

13 2

17

2

69696

X

Y

301

x

n3

4817

y

n2

,

19861

y

n2

1241

x

n3

14 2

17

2

64

X

Y

19861

x

n3

4817

y

n3

,

19861

y

n3

81889

x

n3

15 2

17

2

1183744

X

(7)

B. 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.NO Parabola (X,Y)

1

4

17

2

64

X

Y

5

x

n1

y

n1

,

5

y

2n2

17

x

2n2

8

2

32

17

2

4096

X

Y

73

x

n1

x

n2

,

5

x

2n3

301

x

2n2

64

3

2112

17

2

17842176

X

Y

4817

x

n1

x

n3

,

5

x

2n4

19861

x

2n2

4224

4

132

17

2

69696

X

Y

301

x

n1

y

n2

,

5

y

2n3

1241

x

2n2

264

5

8708

17

2

303317056

X

Y

19861

x

n1

y

n3

,

5

y

2n4

81889

x

2n2

17416

6

132

17

2

69696

X

Y

5

x

n2

73

y

n1

,

301

y

2n2

17

x

2n3

264

7

8708

17

2

303317056

X

Y

5

x

n3

4817

y

n1

,

19861

y

2n2

17

x

2n4

17416

8

544

17

2

1183744

X

Y

5

y

n2

301

y

n1

,

1241

y

2n2

17

y

2n3

1088

9

35904

17

2

5156388864

X

Y

5

y

n3

19861

y

n1

,

81889

y

2n2

17

y

2n4

71808

10

32

17

2

4096

X

Y

4817

x

n2

73

x

n3

,

301

x

2n4

19861

x

2n3

64

11

4

17

2

64

X

Y

301

x

n2

73

y

n2

,

301

y

2n3

1241

x

2n3

8

12

132

17

2

69696

X

Y

19861

x

n2

73

y

n3

,

301

y

2n4

81889

x

2n3

264

13

132

17

2

69696

X

Y

301

x

n3

4817

y

n2

,

19861

y

2n3

1241

x

2n4

264

14

4

17

2

64

X

Y

19861

x

n3

4817

y

n3

,

19861

y

2n4

81889

x

2n4

8

15

544

17

2

1183744

X

Y

301

y

n3

19861

y

n2

,

81889

y

2n3

1241

y

2n4

1088

G. Some Special Cases Of The Solutions Are Given Below

1)

P

y10

t

3,x1

2

153

P

x6

 

t

3,y 2

8

 

t

3,y 2

t

3,x1

2

2)

9

P

y6

 

t

3,x 2

17

P

x10

t

3,y1

2

8

 

t

3,x 2

t

3,y1

2

3)

P

y10

t

3,2x2

2

17

6

P

x41

2

 

t

3,y 2

8

 

t

3,y 2

t

3,2x2

2

4)

36

P

y81

 

t

3,x 2

17

P

x10

t

3,2y2

2

8

 

t

3,x 2

t

3,2y2

2

5)

9

P

y6

t

3,2x2

2

17

36

P

x81

t

3,y1

2

8

t

3,2x2

2

t

3,y1

2

6)

6

P

y41

2

t

3,x1

2

17

3

P

x3

2

t

3,2y2

2

8

t

3,x1

2

t

3,2y2

2

III.CONCLUSIONS

(8)

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.Gopolan, s.vidhyalakshmi and A.Kavitha, on the integral solutions of the Binary quadratic Equation

X

2

4

K

2

1

Y

2

4

t, Bulletin of Mathematics and Statistics Research, 2(1)(2014)42-46.

[6] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, On the binary quadratic Diophantine equation

x

2

3

xy

y

2

33

x

0

, International Journal of Scientific Engineering and Applied Science, 1(4) (2015) 222-225.

[7] 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. [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) (2013) 22-24

[9] A.Kavitha and P.Lavanya, on the positive Pell equation

y

2

20

x

2

16

to appear in International journal of Emerging Technologies in Engineering Research,5(3) (2017).

Figure

Table 1: Examples
Table 2: Hyperbola

References

Related documents

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

x 2  6 2  3 As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of Pell Equations and determine their

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

As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of positive pell equations and determine their

As the binary quadratic Diophantine equation are rich in variety, one may search for the other choices of negative Pell equations and determine their integer

As the binary quadratic Diophantine equation are rich in variety, one may search for the other choices of positive pell equations and determine their integer

The wind turbine was created to leverage this free energy resource by converting the kinetic energy in the wind into mechanical power that runs a generator to

Neoplasms Cardiovascular and circulatory diseases Chronic respiratory diseases Cirrhosis of the liver Digestive diseases (except cirrhosis) Neurological disorders Mental and