On The Negative Pell Equation

International Journal of Research & Review (www.gkpublication.in) 183 Vol.2; Issue: 4; April 2015

International Journal of Research and Review

www.ijrrjournal.com E-ISSN: 2349-9788; P-ISSN: 2454-2237 Short Communication

On The Negative Pell Equation

2 19 2 35x

y  

M.A. Gopalan¹, S.Vidhyalakshmi¹, T.R. Usharani², M. Arulmozhi³

1Professor, ²Lecturer, ³PG Student,

Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India.

Corresponding author: M.A. Gopalan

Received: 11/03/2015 Revised: 02/04/2015 Accepted: 04/04/2015

ABSTRACT

The negative Pell equation represented by the binary quadratic equation _y² _35x²_19 is analyzed for its non-zero distinct integer solutions. A few interesting relations among the solutions are presented. Employing the solutions of the equation under consideration, the integer solutions for a few choices of hyperbola and parabola are obtained.

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

2010 Mathematics subject classification:11D09

INTRODUCTION Diophantine equation of the form

2 1

y2 Dx  where D is a given positive square-free integer, is known as Pell equation and is one of the oldest Diophantine equation that has interested mathematicians all over the world, since antiquity, J.L. Lagrange proved that the positive Pell equation _y² _{ Dx}² _1has infinitely many distinct integer solutions where as the negative Pell equation

2 1

y2 Dx  does not always have a solution. In, ^[1] an elementary proof of a criterion 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

4 7 , 2 1 2 3

y  x  y²  x²  have no integer solutions, whereas _y²₆₅x²_1,y² ₂₀₂x²₁ have integer solutions. In this context, one may refer. ^[2-9] 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 _y² _35x² _19 is considered and infinitely many integer solutions are obtained. A few interesting relations among the solutions are presented.

METHODS OF ANALYSIS

International Journal of Research & Review (www.gkpublication.in) 184 Vol.2; Issue: 4; April 2015

The negative Pell equation representing hyperbola under consideration is

2 19 2 35

y  x 

(1)

whose smallest positive integer solution is 0 1

x , 4

y0 

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

y2 35x2 1 whose general solution is given by

fs ys

gs

2

~ 1 , 35 2

1 xs

~  

Where

) 1 35 6 1 ( ) 35 6 s (

f    



 s s

) 1

35 6 1 ( ) 35 6 s (

g    



 s s

, s=

0, 1, 2, 3,...

Applying Brahmagupta lemma between )

(x_0,y₀ and (x~_s,~y_s)the other integer solutions of (1) are given by

gs fs

35 4 1

2xs   ,

gs fs 35 1 4

2ys  

The recurrence relations satisfied by x and y are given by

1 59 , 0 4 , 1 0 12 2

3

ys      y  y 

ys ys

1 10 , 0 1 , 1 0 12 2

3

xs      x  x  xs

xs

Some numerical examples of x and y satisfying (1) are given in the following table

Table 1: Numerical Examples

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

1. xs is alternatively odd and even 2. ys is alternatively even and odd

3. 0(mod2)

1 - x2s 

4. 0(mod4) y2s 

A few interesting properties between the solutions and special numbers are given below:

1. 38]

2 8 2

2 70 2

19[

6   y s 

x s is a

nasty number

2. ]

1 ) 35 6 ( 1 ) 35 6 [(

3 3] 8 3 2 70 3 [ 19

1 



 



 

 

s s

y s x s

is a cubical integer

3. Each of the following properties represents the perfect square

 [70 8 ] 2

19 1

2 2 2

2_s  y _s  x

 [118 8 ] 2

19 1

3 2 2

2_s  x _s  x

 [352 2 ] 2

57 1

4 2 2

2_s  x _s  x

 ] 2

35 35 8

2 19 [

35

2 2 2

2_s  y _s 

x

 ] 2

35 35 2

10 57 [

35

3 2 2

2_s  y _s 

x

 ] 2

35 35 8

238 1349[

35

4 2 2

2_s  y _s 

x

 [1408 118 ] 2

19 1

4 2 3

2_s  x _s  x

 ] 2

35 35 59

57 [ 35

2 2 3

2_s  y _s 

x

 ] 2

35 35 118

20 19 [

35

3 2 3

2_s  y _s 

x

 ] 2

35 35 59

119 57 [

35

4 2 3

2_s  y _s 

x

s xs

ys

0 1 4

1 10 59

2 119 704

3 1418 8389

4 16897 99964

5 201346 1191179

6 2399255 14194184

7 28589714 169139029

International Journal of Research & Review (www.gkpublication.in) 185 Vol.2; Issue: 4; April 2015

 ] 2

35 35 1408

2 1349[

35

2 2 4

2_s  y _s 

x

 ] 2

35 35 704

10 57 [

35

3 2 4

2_s  y _s 

x

 ] 2

35 35 1408

238 19 [

35

4 2 4

2_s  y _s 

x

 [2 20 ] 2

19 1

2 2 3

2_s  y _s  y

 [ 119 ] 2

114 1

2 2 4

2_s  y _s  y

 [20 238 ] 2

19 1

3 2 4

2_s  y _s  y

4. 38 1

456 2 3

38xs    xs xs

5. 228 1

38 2 1

38ys    xs xs

6. 38 1

228 2 3

38ys    xs xs

7. 228 1

2698 2 3

38ys    xs xs

8. 1444x_s3 17328y_s1 102524x_s1 9. 8664x_s3 1444y_s1 102524x_s2 10. 38x_s3x_s1456y_s1x_s12698x_s²_1 11. 8664x_s3x_s21444y_s1x_s2 102524x_s²_2 12. 8664x_s3 17328y_s2 8664x_s1

13. 38x_s3 38y_s2 228x_s2 14. 38x_s3x_s176y_s2x_s1 38x_s²_1 15. 38x_s3x_s2 38y_s2x_s2 228x_s²_2 16. 102524x_s3 17328y_s3 1444x_s1 17. 8664x_s3 1444y_s3 1444x_s2

18. 102524x_s3x_s117328y_s3x_s1 1444x_s²_1 19. 228x_s3x_s238y_s3x_s2 38x_s²_2

20. 1444y_s2 8664y_s1 50540x_s1 21. 8664y_s2 1444y_s1 50540x_s2

22. 38y_s2x_s1228y_s1x_s1 1330x_s²_1 23. 228y_s2x_s2 38y_s1x_s2 1330x_s²_2 24. 1444y_s3 102524y_s1 606480x_s1 25. 38y_s3 38y_s1 2660x_s2

26. 1444y_s3x_s1102524y_s1x_s1 606480x_s²_1 27. 38y_s3x_s238y_s1x_s2 2660x_s²_2 28. 8664y_s3102524y_s2 50540x_s1 29. 1444y_s3x_s1102524y_s2x_s150540x_s²_1 30. 8664y_s3x_s1102524y_s2x_s1 50540x_s²_1

31. 38y_s3x_s2228y_s2x_s2 1330x_s²_2 REMARKABLE OBSERVATIONS

1. Let N be any non-zero positive integer such that

2 1 N y^{2n 1} 

 ^

It is seen that 8t_3,N 1 y₂²_n1 Similarly 8t_3,M 1 x₂²_n

2. Let m and n be two non-zero distinct positive integer such that

n

n n x

y 



x 2 ,

m _n

Note that m>n>0 treat m, n as the generators of the Pythagorean triangle T(α,β,γ) .

Where α=2mn, β=m²-n² and γ=m²+n²

Let A, P represent the area and perimeter of T(α,β,γ). Then the following interesting relations are observed:

(1). -70 69 40

(2). 4

P 4A - 70 -

71  

(3). 4

P 280A 71

-  



International Journal of Research & Review (www.gkpublication.in) 186 Vol.2; Issue: 4; April 2015

Each of the following represents the hyperbola:

S. No Hyperbola ( y)

gs ), x s(

f  

1 x² 140y² 1444

] 8 70

19[ 1

1

1 

  _s

s y

x ^, [ 4 ]

19 35 2

1

1 

  _s

s x

y

2 x² 35y² 1444

] 8 118

19[ 1

2

1 

  _s

s x

x ^, [2 20 ]

19 35

1

2 

  _s

s x

x

3 4x² 35y² 51984

] 2 352

57[ 1

3

1 

  _s

s x

x , [ 119 ]

144 35

1

3 

  _s

s x

x

4 35x² 35y² 1444

] 35 35 8

2 19 [

35

1

1 

  _s

s y

x , [2 8 ]

19 35

1

1 

  _s

s x

y

5 35x² 35y² 12996

] 35 35 2

10 57 [

35

2

1 

  _s

s y

x , [ 59 ]

57 35

1

2 

  _s

s x

y

6 35x² 35y² 7279204

] 35 35 8

238 1349[

35

3

1 

  _s

s y

x , [2 1408 ]

1349 35

1

3 

  _s

s x

y

7 x² 35y 1444

] 118 1408

19[ 1

3

2 

  _s

s x

x ^, [20 238 ]

19 35

2

3 

  _s

s x

x

8 35x² 35y² 12996

] 35 35 59

57 [ 35

1

2 

  _s

s y

x , [10 4 ]

57 35

2

1 

  _s

s x

y

9 35x² 35y² 1444

35 ] 35 118

20 19 [

35

2

2 

  _s

s y

x , [20 118 ]

19 35

2

2 

  _s

s x

y

10 35x² 35y² 12996

] 35 35 59

119 57 [

35

3

2 

  _s

s y

x [10 704 ]

57 35

2

3 

  _s

s x

y

11 35x² 35y² 7279204

] 35 35 1408

2 1349[

35

1

3 

  _s

s y

x , [238 8 ]

1349 35

3

1 

  _s

s x

y

12 35x² 35y² 12996

] 35 35 704

10 57 [

35

2

3 

  _s

s y

x , [119 59 ]

57 35

3

2 

  _s

s x

y

13 35x² 35y² 1444

3] 35 1408 35 3

238 [ 19

35

 

 ys

xs , [238 1408 ]

19 35

3

3 

  _s

s x

y

14 35x² 35y² 50540

] 20 2

19[ 1

1

2 

  _s

s y

y ^{, [118} ₁ ⁸ ₂^]

35 19

1

 

 ys ys

15 35x² y² 1819440 [ 119 ]

114 1

1

3 

  _s

s y

y , [1408 118 ]

35 19

1

3

2 

  _s

s y

y

16 35x² 35y² 50540 [20 238 ]

19 1

2

3 

  _s

s y

y , [1408 118 ]

35 19

1

3

2 

  _s

s y

y

International Journal of Research & Review (www.gkpublication.in) 187 Vol.2; Issue: 4; April 2015

Each of the following represents the parabola

S. No Parabola

) y s( g ), x (

f_s²  

1 140y² 19x1444

2 ] 8 70

19[ 1

2 2 2

2_s  y _s 

x ^, [ 4 ]

19 35 2

1

1 

  _s

s x

y

2 35y² 19x1444

2 ] 8 118

19[ 1

3 2 2

2_s  x _s 

x ^, [2 20 ]

19 35

1

2 

  _s

s x

x

3 35y² 228x51984

2 ] 2 352

57[ 1

4 2 2

2_s  x _s 

x ^, [ 119 ]

144 35

1

3 

  _s

s x

x

4 35y² 19x1444

2 ] 35 35 8

2 19 [

35

2 2 2

2_s  y _s 

x , [2 8 ]

19 35

1

1 

  _s

s x

y

5 35y² 57x12996

2 3] 35 2 2 2 35 2 10 [ 57

35 x s  y s  , [ 2 59 1]

57

35 ys  xs

6 35y² 1349x7279204

2 4] 35 2 8 2 35 2 238 [ 1349

35 x s  y s  , [2 3 1408 1] 1349

35

 

 xs

ys

7 35y 19x1444

2 ] 118 1408

19[ 1

4 2 3

2_s  x _s 

x ^, [20 238 ]

19 35

2

3 

  _s

s x

x

8 35y² 57x12996

2 ] 35 35 59

57 [ 35

2 2 3

2_s  y _s 

x , [10 4 ]

57 35

2

1 

  _s

s x

y

9 35y² 19x1444

2 3] 35 2 118 3 35 2 20 [ 19

35 x s  y s  , [20 2 118 2] 19

35

 

 xs

ys 10 35y² 57x12996

2 4] 35 2 59 3 35 2 119 [ 57

35 x s  y s  , [10 3 704 2] 57

35

 

 xs

ys 11 35y² 1349x7279204

2 2] 35 2 1408 4 35 2 2 [ 1349

35 x s  y s  , [238 1 8 3] 1349

35

 

 xs ys

12 35y² 57x12996

2 3] 35 2 704 4 35 2 10 [ 57

35 x s  y s  , [119 2 59 3] 57

35

 

 xs ys

13 35y² 19x1444

2 4] 35 2 1408 4 35 2 238 [ 19

35 x s  y s  , [238 3 1408 3] 19

35

 

 xs

ys 14 y² 665x50540

2 2] 20 2 3 2 2 [ 19

1 y s  y s  , [118 1 8 2] 35

19 1

 

 ys ys

15 y² 3990x1819440

2 2] 119 2 4 [ 2 114

1 y s  y s  , [1408 2 118 3] 35

19 1

 

 ys

ys 16 y² 665x50540

2 3] 238 2 4 20 2 [ 19

1 y s  y s  , [1408 2 118 3] 35

19 1

 

 ys

ys

CONCLUSION

In this paper, We have presented infinitely many integer solutions for the

hyperbola represented by the negative Pell equation _y² _35x²_19. As the binary

International Journal of Research & Review (www.gkpublication.in) 188 Vol.2; Issue: 4; April 2015

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

REFERENCES

1. R.A.Mollin and Anitha srinivasan, ‟ A Note on the negative pell equation ”, International Journal of Algebra, 2010.Vol 4,no.19,919-922.

2. E.E Whitford, ‟Some solutions of the pellian Equationsx²Ay²4^” JSTOR: Annals of Mathematics, Second series, (1913-1914). vol.15,no

4 1,(157- 160) .

3. Ahmet Tekcan , Betw Gezer and Osman Bizim, ‟ On the integer solutions of the pell Equation x²dy²2^{t ”,World} Academy of science, Engineering and Technology 2007,1, (522-526).

4. Ahmet Tekcan, ‟ The pell equation y t

k

k ) 2

(

x² ² ²  ^{”, World}

Academy of science, Engineering and Technology, 2008,19,(697-701).

5. Merve Guney, ‟ Solutions of the pell equations , x²(a²b²2b)y²2^t when N(1,4)”,Mathematica Aeterna , 2012, Vol 2, no.7 (629-638) . 6. M.A. Gopalan, S. vidhyalakshmi, N.

Thiruniraiselvi,” A study on the hyperbola y² 8x²31,”International Journal of latest Research in science and technology.” 2013, 2(1), 454-456.

7. V.Sangeetha, M.A.Gopalan and Manju Somanath, ‟On the integral solutions of the pell equation x²13y²3^{t ”,} International journal of appield Mathematical research, 2014,Vol .3, issue 1 (58-61).

8. M.A.Gopalan,G.sumathi,S.vidhyalaksh mi, ‟ Observations on the hyperbola

y 3t

19

x²  ² ”,Scholars Journal of the Engineering and Technology,(2014), Vol:2(2A):152-155.

9. M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, ‟ On the integral solution of the Binary quadratic equation

t 2

2 15y 11

x   ”, Scholars Journal of the Engineering and Technology, 2014,Vol 2(2A),156-158.

**************

How to cite this article: Gopalan MA, Vidhyalakshmi S, Usharani TR et. al. On the negative Pell equationy2 35x219. Int J Res Rev. 2015; 2(4):183-188.