Observation on the Negative Pell Equation

Observation on the Negative Pell Equation

23

12

2

2





x

y

M. A. Gopalan1, A. Sathya2, S. Nivetha3*

1

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

2

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

Tamil nadu, India.

3

PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India.

Abstract: The binary quadratic equation represented by the negative pellian 2 12 2 23



 x

y is analyzed for its distinct integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas.

Keywords:Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions

I. INTRODUCTION

The subject of Diophantine equation is one of the areas in Number Theory that has attracted many Mathematicians since antiquity and it has a long history. Obviously, the Diophantine equation are rich in variety [1-3]. In particular, the binary quadratic

diophantine equation of the form y2 Dx2 N



N 0, D 0andsquarefree



  

 is referred as the negative form of the pell equation

(or) related pell equation. It is worth to observe that the negative pell equation is not always solvable. For example, the

equations 2 3 2 1

  x

y , 2 7 2 4

  x

y have no integer solutions whereas 2 65 2 1

  x

y , 2 202 2 1

  x

y have integer solutions. In this

context, one may refer [4-10] for a few negative pell equations with integer solutions. In this communication, the negative pell

equation given by y212x223is considered and analysed for its integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas.

II. METHODOFANALYSIS The Negative Pell equation representing hyperbola under consideration is

12 23

2 2

  x

y (1)

The smallest positive integer solutions of (1) are

x02,y05

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

12 1 2 2

  x

y (2)

whose initial solution is given by

2 ~

0

x ,~y07

The general solution



x~n,y~n



of (2) is given by

n

n g

x

3 4

1

~ _{ ,}

n

n f

y

2 1

~ _

where

1 1

) 3 4 7 ( ) 3 4 7

(     

 n n

n

f

(7 4 3) (7 4 3) , 1,0,1,.... 1

1

  

 

   n

g_n n n

n n

n

f

g

x

3

4

5

1







n n

n f g

y 2 3

2 5 1 



The recurrence relation satisfied by the solution x and y are given by

xn314xn2xn10

yn314yn2yn10

Some numerical examples of x_nand y_n satisfying (1) are given in the Table: 1 below:

Table: 1 Numerical Examples

n

n

x

y

_n

0 2 5

1 24 83

2 334 1157

3 4652 16115

4 64794 224453

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

1) xnvalues are even.

2) ynvalues are odd .

A. Relation Among The Solutions Are Given Below 1) xn197xn328yn3

2) xn27xn12yn1

3) 7xn397xn22yn1

4) 7yn224xn2yn1

5) xn397xn128yn1

6) yn197yn3336xn3

7) yn224xn17yn1

8) yn3336xn197yn1

9) 48xn2 yn3yn1

10) 14yn2yn3yn1

11) xn114xn2xn3

12) xn17xn22yn2

13) 7xn197xn22yn3

14) 7xn3xn22yn3

15) 7yn2 yn324xn2

16) xn1xn34yn2

17) 7yn197yn224xn3

18)

7

x

_n2



x

_n3



2

y

_n2 19) 24xn17yn397yn2

B. Each Of The Following Expressions Represents A Nasty Number

1)



48 10 46



23 6

2 2 2

2n  yn 

x

2)



83 5 46



23 6

3 2 2

2n  xn 

x

3)



1157 5 644



161 3

4 2 2

2n  xn 

x

4)



288 5 161



161 12

3 2 2

2n  yn 

x

5)



8016 10 4462



2231 6

4 2 2

2n  yn 

x

6)



48 166 322



161 6

2 2 3

2n  y n 

x

7)



24 1157 2231



2231 12

2 2 4

2n  yn 

x

8)



12 144 276



23 1

2 2 3

2n  yn 

y

9)



167 322



161 6

2 2 4

2n  yn 

y

10)



1157 83 46



23 6

4 2 3

2n  xn 

x

11)



288 83 23



23 12

3 2 3

2n  yn 

x

12)



4008 83 161



161 12

4 2 3

2n  yn 

x

13)



288 1157 161



161 12

3 2 4

2n  yn 

x

14)



4008 1157 23



23 12

4 2 4

2n  yn 

x

15)



144 2004 276



23 1

3 2 4

2n  yn 

y

C. Each Of The Following Expressions Represents A Cubical Integer

1)



48 3 3 10 3 3 144 1 30 1



23 1

 



  n  n  n

n y x y

x

2)



83 3 3 5 3 4 249 1 15 2



23 1

  

  n  n  n

n x x x

x

3)



1157 3 3 5 3 5 3471 1 15 3



322 1

  

  n  n  n

n x x x

x

4)



288 _{3 3} 5 _{3 4} 864 ₁ 15 ₂



161 2

  

  n  n  n

n y x y

x

5)



8016 3 3 10 3 5 24048 1 30 3



2231 1

 



  n  n  n

n y x y

x

6)



48 3 4 166 3 3 144 2 498 1



161 1

 



  n  n  n

n y x y

7)



24 _{3 5} 1157 _{3 3} 72 ₃ 3471 ₁



2231 2   

  n  n  n

n y x y

x

8)



2 3 4 24 3 3 6 2 72 1



23

1

  

  n  n  n n y y y

y

9)



3 4 167 3 3 3 3 501 1



161 1

 



  n  n  n n y y y

y

10)



1157 3 4 83 3 5 3471 2 249 3



23 1

 



  n  n  n

n x x x

x

11)



288 3 4 83 3 4 864 2 249 2



23 2

 



  n  n  n

n y x y

x

12)



4008 3 4 83 3 5 12024 2 249 3



161 2

 



  n  n  n

n y x y

x

13)



288 3 5\ 1157 3 4 864 3 3471 2



161 2

 



  n  n  n

n y x y

x

14)



4008 3 5\ 1157 3 5 12024 3 3471 3



23 2

 



  n  n  n

n y x y

x

15)



12 3 5\ 167 3 4 36 3 501 2



23

2

 



  n  n  n n y y y

y

D. Each Of The Following Expressions Represents A Biquadratic Integer

1)



48 10 192 40 138



23 1 2 2 2 2 4 4 4

4n  yn  xn  yn 

x

2)



83 5 20 332 138



23 1 2 2 3 2 5 4 4

4n  xn  xn  xn 

x

3)



1157 5 20 4628 1932



322 1 2 2 4 2 6 4 4

4n  xn  xn  xn 

4)



288 5 20 1152 966



161 2 2 2 3 2 5 4 4

4n  yn  yn  xn 

x

5)



8016 10 40 32064 13386



2231 1 2 2 4 2 6 4 4

4n  yn  yn  xn 

x

6)



48 166 664 192 966



161 1 3 2 2 2 4 4 5

4n  yn  yn  xn 

x

7)



24 1157 4628 96 6693



2231 2 4 2 2 2 4 4 6

4n  yn  y n  xn 

x

8)



12 48 4 69



23 2 3 2 2 2 4 4 5

4n  yn  yn  yn 

y

9)



167 668 4 966



161 1 4 2 2 2 4 4 6

4n  yn  yn  yn 

y

10)



1157 83 332 4628 138



23 1 3 2 4 2 6 4 5

4n  xn  xn  xn 

x

11)



288 83 332 1152 69



23 2 3 2 3 2 5 4 5

4n  yn  yn  xn 

x

12)



4008 83 332 16032 483



161 2 3 2 4 2 6 4 5

4n  yn  yn  xn 

x

13)



288 1157 4628 1152 483



161 2 4 2 3 2 5 4 6

4n  yn  yn  xn 

14)



4008 1157 4628 16032 69



23

2

4 2 4

2 6

4 6

4n  yn  yn  xn 

x

15)



12 167 668 48 69



23 2

4 2 3 2 5 4 6

4n  yn  yn  yn 

y

E. Each Of The Following Expressions Represents A Quintic Integer

1)



48 5 5 10 5 5 240 3 3 50 3 3 480 1 100 1



23 1

 

 



  n  n  n  n  n

n y x y x y

x

2)



83 _{5 5} 5 _{5 6} 415 _{3 3} 25 _{3 4} 830 ₁ 50 ₂



23 1

 

 



  n  n  n  n  n

n x x x x x

x

3)



1157 5 5 5 5 7 5785 3 3 25 3 5 11570 1 50 3



322 1

 

 



  n  n  n  n  n

n x x x x x

x

4)



288 5 5 5 5 6 1440 3 3 25 3 4 2880 1 50 2



161 2

 

 



  n  n  n  n  n

n y x y x y

x

5)



8016 _{5 5} 10 _{5 7} 40080 _{3 3} 50 _{3 5} 80160 ₁ 100 ₃



2231 1

 

 



  n  n  n  n  n

n y x y x y

x

6)



48 5 6 166 5 5 240 3 4 830 3 3 480 2 1660 1



161 1

 

 



  n  n  n  n  n

n y x y x y

x

7)



24 5 7 1157 5 5 120 3 5 5785 3 3 240 3 11570 1



2231 2

 

 



  n  n  n  n  n

n y x y x y

x

8)



5 6 12 5 5 5 3 4 60 3 3 10 2 120 1



23

2

 

  

  n  n  n  n  n n y y y y y

y

9)



5 7 167 5 5 5 3 5 835 3 3 10 3 1670 1



161

1

 

 



  n  n  n  n  n n y y y y y

y

10)



1157 5 6 83 5 7 5785 3 4 415 3 5 11570 2 830 3



23 1

 

 



  n  n  n  n  n

n x x x x x

x

11)



288 5 6 83 5 6 1440 3 4 415 3 4 2880 2 830 2



23 2

 

 



  n  n  n  n  n

n y x y x y

x

12)



4008 5 6 83 5 7 20040 3 4 415 3 5 40080 2 830 3



161 2

 

 



  n  n  n  n  n

n y x y x y

x

13)



288 5 7 1157 5 6 1440 3 5 5785 3 4 2880 3 11570 2



161 2

 

 



  n  n  n  n  n

n y x y x y

x

14)



4008 _{5 7} 1157 _{5 7} 20040 _{3 5} 5785 _{3 5} 40080 ₃ 11570 ₃



23 2

 

 



  n  n  n  n  n

n y x y x y

x

15)



12

_{5 7}

167

_{5 6}

60

_{4 6}

835

_{4 5}

120

₃

1670

₂



23

2

 

 







n



n



n



n



n

n

y

y

y

y

y

III.REMARKABLEOBSERVATIONS

1) 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 Hyperbolas

S.NO Hyperbola (X,Y)

1 _Y2_12X2 _2116





1 1

1

1 10 ,48 10 4yn  xn xn  yn

2 2 ₁₂ 2 ₂₁₁₆



 X

Y



2xn224xn1,83xn15xn2



3 _Y2_12X2_414736





3 1 1

3 334 ,1157 5

2xn  xn xn  xn

4 _Y2_3X2 _25921





2 1 1

2 166 ,288 5

4yn  xn xn  yn

5 2 ₁₂ 2 _19909444



 X

Y



2314xn14yn3,8016xn110yn3



6 _Y2_12X2_103684





1 2

2

1 10 ,48 166

48yn  xn xn  yn

7 _Y2_12X2_4977361





3 1

3

1 5 ,1157 24

334yn  xn yn  xn

8 2_3 2 _76176

X

Y



83yn15yn2,12yn2144yn1



9 _Y2_3X2 _14930496





1 3

3

1 5 ,12 2004

1157yn  yn yn  yn

10 2 ₂₄ 2 ₂₁₁₆



 X

Y



12xn3167xn2 ,1157xn283xn3



11 _Y2_12X2_529





2 2

2

2 83 ,288 83

24yn  xn xn  yn

12 2 ₁₂ 2 ₂₅₉₂₁



 X

Y



24yn31157xn2 ,4008xn283yn3



13 _Y2_12X2 _25921





2 3

3

2 83 ,288 1157

334yn  xn xn  yn

14 2 ₁₂ 2 ₅₂₉



 X

Y



334yn31157xn3,4008xn31157yn3



15 2_3 2 _76176

X

Y



1157yn283yn3 ,144yn32004yn2



2) Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabolas

which are presented in Table: 3 below:

Table: 3 Parabolas

S.NO Parabola (X,Y)

1 _{23 12} 2 ₂₁₁₆

  X

Y



4yn110xn1 ,48x2n210y2n246



2 _{23 12} 2 ₂₁₁₆

  X

Y



2xn124xn1,83x2n25x2n346



3 _{322 12} 2 ₄₁₄₇₃₆

  X

Y



2xn3334xn1,1157x2n25x2n4644



4 _{322 12} 2 ₁₀₃₆₈₄

  X

Y



4yn2166xn1,288x2n25y2n3161



5 _{2231 12} 2 _19909444

  X

Y



2314xn14yn3,8016y2n210x2n44462



6 _{161 12} 2 ₁₀₃₆₈₄

  X

Y



48yn110xn2 ,48x2n3166y2n2322



7 _{4462 48} 2 _19909444

  X

Y



334yn15xn3,24x2n41157y2n22231



8 ₄₆ 2 ₂₅₃₉₂

 X

Y



83yn15yn2,12y2n3144y2n2276



9 ₆₄₄ 2 _4976832

 X

Y



1157yn15yn3,12y2n42004y2n23864



10

2116 24

23Y X2 



12xn3167xn2,1157x2n383x2n446



11 _{23 24} 2 ₁₀₅₈



 X

Y



24yn283xn2,288x2n383y2n323



12 _{161 24} 2 ₅₁₈₄₂



 X

Y



24yn31157xn2,4008x2n383y2n4161



13 _{161 24} 2 ₅₁₈₄₂



 X

Y



334yn283xn3,288x2n41157y2n3161



14 _23Y24X2_1058



334 1157 ,4008 1157 23



4 2 4

2 3

3     

 n n n

n x x y

y

15 _46YX2 _25392



_{1157 83 ,144 2004 276}



3 2 4

2 3

2     

 n n n

n y y y

REFERENCES

[1] L.E. Dickson., History of Theory of numbers, Chelsea Publishing Company, volume-II, Newyork, 1952.

[2] L.J. Mordell, Diophantine Equations, Academic Press, Newyork, 1969.

[3] S.J. Telang., Number Theory, Tata Mc Graw Hill Publishing Company Limited, New Delhi, 2000.

[4] M.A. Gopalan, S. Vidhyalakshmi, J. Shanthi, and D.Kanaka, “On the Negative Pell Equation 2 15 2 6

  x

y ,” Scholars Journal of Physics, Mathematics and

statistics, vol. 2, Issue-2A (Mar-May), pp 123-128, 2015.

[5] M.A. Gopalan, S. Vidhyalakshmi, T. R. Usharani, and M. Arulmozhi, “On the Negative Pell Equation 2 35 2 19

  x

y ,” International Journal of Research and

Review, vol. 2, Issue-4 April 2015, pp 183-188.

[6] M.A. Gopalan, V. Geetha, S.Sumithira, “Observations on the Hyperbola 2 55 2 6

  x

y ,” International Research Journal of Engineering and Technology, vol.

2, Issue-4 July 2015, pp 1962-1964.

[7] K. Meena, M.A. Gopalan, T.Swetha, “On the Negative Pell Equation 2 40 2 4

  x

y ,” International Journal of Emerging Technologies in Engineering

Research, vol. 5, Issue-1, January (2017), pp 6-11.

[8] K. Meena, S. Vidhyalakshmi and G. Dhanalakshmi, “On the Negative Pell Equation 2 5 2 4

  x

y ,” Asian Journal of Applied Science and Technology, vol. 1,

Issue-7, August 2017, pp 98-103.

[9] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, “On the Negative Pellian Equation 2 20 2 31

  x

y ,” International Journal of Pure and Applied

Mathematics, vol. 119 No. 14, 2018, pp 199-204.

[10] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, “Observations on the Positive Pell Equation 2 20



2 1



  x

y ,” International Journal of Pure and Applied