Observation on the Negative Pell Equation

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Observation on the Negative Pell Equation

36

40

2

2





x

y

K. Meena1, S. Vidhyalakshmi2, J. Srilekha3*

1

Former VC, Bharathidasan University, Trichy-620 024, Tamil nadu, India.

2

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

3

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

Abstract: The binary quadratic equation represented by the negative pellian _y2_40_x2_36_{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. Also, the relations between the solutions and special figurate numbers are exhibited.

Keywords: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions and Figurate numbers.

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 2 40 2 36



 x

y is 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. Also, the relations between the solutions and special figurate numbers are exhibited.

II. NOTATIONS

1)







_

  



  

 

2 2 1 1

,

m n n

tmn - Polygonal number of rank n with size m

2) pm n



n

 



m



n



m





n  1 2  5

6 1

- Pyramidal number of rank n with size m

III.METHODOFANALYSIS

The negative Pell equation representing hyperbola under consideration is 36

40 2 2



 x

y (1)

whose smallest positive integer solution is 2 , 1 0

0 y 

x

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

1 40 2

2



 x

y

whose general solution is given by

n n n

n g y f

x

2 1 ~ , 10 4

1

~ _{ }

where





1





1

40 3 19 40

3

19    

 n n

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

193 40



1



193 40



1 , 1,0,1,2...

   n

g_n n n

Applying Brahmagupta lemma between



x0,y0



and



x~n,~yn



, the other integer solutions of (1) are given by

n n

n f g

x

10 2

1

2 1

1 



n n

n f g

y₁  10

The recurrence relations satisfied by x and y are given by 0

38 2 1

3    

 n n

n x x

x

0

38 _{2 1}

3    

 n n

n y y

y

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

Table: 1 Numerical examples

n

1

 n

x yn1

-1 1 2

0 25 158

1 949 6002

2 36037 227918

3 1368457 8654882

4 51965329 328657598

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

A. xn1 is always odd, yn1is always even.

B. Relations Among the Solutions 1) xn219xn13yn10

2) 27yn2171xn29xn10

3) 3yn3721xn219xn10

4) xn338xn2xn10

5) 19xn3721xn23yn10

6) xn₃19xn₂3yn₂0

7) 19xn3xn23yn30

8) xn₃114yn₁721xn₁0

9) xn36yn2xn10

10) 721xn3xn1114yn30

11) yn2120xn119yn10

12) yn34560xn1721yn10

13) 19yn3721yn2120xn10

14) 120xn219yn2yn10

15) yn3240xn2yn10

16) yn3120xn219yn20

17) 120xn3721yn219yn10

18) 721yn34560xn3yn10

19) 120xn319yn3yn20

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C. Each Of The Following Expressions Represents A Nasty Number

1)



79 54



9 2

3 2 2

2n xn  x

2)



3001 79 54



9 2

4 2 3

2n  xn  x

3)



3001 2052



171 1

4 2 2

2n xn  x

4)



20 18



3 2

2 2 2

2n yn  x

5)



500 342



57 2

3 2 2

2n yn  x

6)



18980 12978



2163 2

4 2 2

2n yn  x

7)



20 79 342



57 2

2 2 3

2n  yn  x

8)



500 79 18



3 2

3 2 3

2n  yn  x

9)



18980 79 342



57 2

4 2 3

2n  y n  x

10)



20 3001 12978



2163 2

2 2 4

2n  yn  x

11)



500 3001 342



57 2

3 2 4

2n  yn  x

12)



18980 3001 18



3

2

4 2 4

2n  yn  x

13)



25 108



9 1

2 2 3

2n  yn 

y

14)



949 4104



342 1

2 2 4

2n  yn  y

15)



25 949 108



9 1

3 2 4

2n  yn  y

D. Each Of The Following Expressions Represents A Cubical Integer

1)



79 3 3 3 4 237 1 3 2



27 1

  

  n  n  n

n x x x

x

2)



3001 3 4 79 3 5 9003 2 237 3



27 1

 



  n  n  n

n x x x

x

3)



3001 3 3 3 5 9003 1 3 3



1026 1

  

  n  n  n

n x x x

x

4)



20 3 3 3 3 60 1 3 1



9 1

  

  n  n  n

n y x y

x

5)



500 3 3 3 4 1500 1 3 2



171 1

  

  n  n  n

n y x y

x

6)



18980 3 3 3 5 56940 1 3 3



6489 1

  

  n  n  n

n y x y

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7)



20 _{3 4} 79 _{3 3} 60 ₂ 237 ₁



171 1

 



  n  n  n

n y x y

x

8)



500 3 4 79 3 4 1500 2 237 2



9 1

 



  n  n  n

n y x y

x

9)



18980 3 4 79 3 5 56940 2 237 3



171 1

 



  n  n  n

n y x y

x

10)



20 3 5 3001 3 3 60 3 9003 1



6489 1

 



  n  n  n

n y x y

x

11)



500 3 5 3001 3 4 1500 3 9003 2



171 1

 



  n  n  n

n y x y

x

12)



18980 3 5 3001 3 5 56940 3 9003 3



9 1

 



  n  n  n

n y x y

x

13)



3 4 25 3 3 3 2 75 1



54 1

 



  n  n  n

n y y y

y

14)



3 5 949 3 3 3 3 2847 1



2052 1

 



  n  n  n

n y y y

y

15)



25 3 5 949 3 4 75 3 2847 2



54 1

 



  n  n  n

n y y y

y

E. Each Of The Following Expressions Represents A Bi-Quadratic Integer

1)



79 316 4 162



27 1 3 2 2 2 5 4 4

4n xn  xn  xn  x

2)



3001 79 12004 316 162



27 1 4 2 3 2 6 4 5

4n  xn  xn  xn  x

3)



3001 12004 4 6156



1026 1 4 2 2 2 6 4 4

4n xn  xn  xn  x

4)



20 80 4 54



9 1 2 2 2 2 4 4 4

4n yn  xn  yn  x

5)



500 2000 4 1026



171 1 3 2 2 2 5 4 4

4n y n  xn  yn  x

6)



18980 75920 4 38934



6489 1 4 2 2 2 6 4 4

4n yn  xn  y n  x

7)



20 79 80 316 1026



171 1 2 2 3 2 4 4 5

4n  yn  xn  yn  x

8)



500 79 2000 316 54



9 1 2 2 5 4 5

4n  yn  xn  yn  x

9)



18980 79 75920 316 1026



171 1 4 2 3 2 6 4 5

4n  yn  xn  yn  x

10)



20 3001 80 12004 38934



6489 1 2 2 4 2 4 4 6

4n  yn  xn  y n 

x

11)



500 3001 2000 12004 1026



171 1 3 2 4 2 5 4 6

4n  yn  xn  yn 

x

12)



18980 3001 75920 12004 54



9 1 4 2 4 2 6 4 6

4n  yn  xn  yn 

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13)



25 4 100 324



54 1 2 2 3 2 4 4 5

4n  yn  yn  yn  y

14)



949 4 3796 12312



2052 1 2 2 4 2 4 4 6

4n  y n  yn  yn  y

15)



25 949 100 3796 324



54 1 3 2 4 2 5 4 6

4n  yn  yn  yn  y

F. Each Of The Following Expressions Represents A Quintic Integer

1)



79 5 5 5 6 395 3 3 5 3 4 790 1 10 2



27 1     

  n  n  n  n  n

n x x x x x

x

2)



3001 5 6 79 5 7 15005 3 4 395 3 5 30010 2 790 3



27 1     

  n  n  n  n  n

n x x x x x

x

3)



3001 _{5 5 5 7} 15005 _{3 3} 5 _{3 5} 30010 ₁ 10 ₃



1026 1     

  n  n  n  n  n

n x x x x x

x

4)



20 5 5 5 5 100 3 3 5 3 3 200 1 10 1



9 1     

  n  n  n  n  n

n y x y x y

x

5)



500 5 5 5 6 2500 3 3 5 3 4 5000 1 10 2



171 1     

  n  n  n  n  n

n y x y x y

x

6)



18980 5 5 5 7 94900 3 3 5 3 5 189800 1 10 3



6489 1     

  n  n  n  n  n

n y x y x y

x

7)



20 5 6 79 5 5 100 3 4 395 3 3 200 2 790 1



171 1     

  n  n  n  n  n

n y x y x y

x

8)



500 5 6 79 5 6 2500 3 4 395 3 4 5000 2 790 2



9 1     

  n  n  n  n  n

n y x y x y

x

9)



18980 5 6 79 5 7 94900 3 4 395 3 5 189800 2 790 3



171 1     

  n  n  n  n  n

n y x y x y

x

10)



20 5 7 3001 5 5 100 3 5 15005 3 3 200 3 30010 1



6489 1     

  n  n  n  n  n

n y x y x y

x

11)



500 5 7 3001 5 6 2500 3 5 15005 3 4 5000 3 30010 2



171 1     

  n  n  n  n  n

n y x y x y

x

12)



18980 5 7 3001 5 7 94900 3 5 15005 3 5 189800 3 30010 3



9 1     

  n  n  n  n  n

n y x y x y

x

13)



5 6 25 5 5 5 3 4 125 3 3 10 2 250 1



54 1     

  n  n  n  n  n

n y y y y y

y

14)



5 7 949 5 5 5 3 5 4745 3 3 10 3 9490 1



2052 1     

  n  n  n  n  n

n y y y y y

y

15)



25 5 7 949 5 6 125 3 5 4745 3 4 250 3 9490 2



54 1     

  n  n  n  n  n

n y y y y y

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G. Remarkable Observations

1) Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbola which are presented in the Table: 2 below:

Table: 2 Hyperbolas S. No

Hyperbola



X,Y



1 2 ₁₀ 2 ₂₉₁₆



 Y

X



79xn1xn2,xn225xn1



2 2 ₁₀ 2 ₂₉₁₆



 Y

X



3001xn279xn3,25xn3949xn2



3 2 ₁₀ 2 _4210704



 Y

X



3001xn1xn3,xn3949xn1



4 2 ₁₀ 2 ₃₂₄



 Y

X



20xn1yn1,yn12xn1



5 2 ₁₀ 2 ₁₁₆₉₆₄



 Y

X



500xn1yn2,yn2158xn1



6 2 ₁₀ 2 _168428484



 Y

X



18980xn1yn3,yn36002xn1



7 2 ₁₀ 2 ₁₁₆₉₆₄



 Y

X



20xn279yn1,25yn12xn2



8 2 ₁₀ 2 ₃₂₄



 Y

X



500xn279yn2,25yn2158xn2



9 2 ₁₀ 2 ₁₁₆₉₆₄



 Y

X



18980xn279yn3,25yn36002xn2



10 2 ₁₀ 2 _168428484



 Y

X



20xn33001yn1,949yn12xn3



11 2 ₁₀ 2 ₁₁₆₉₆₄



 Y

X



500xn33001yn2,949yn2158xn3



12 2 ₁₀ 2 ₃₂₄



 Y

X



18980xn33001yn3,949yn36002xn3



13 ₁₀ 2 2 ₁₁₆₆₄₀

 Y

X



yn225yn1,79yn1yn2



14 ₁₀ 2 2 _168428160

 Y

X



yn3949yn1,3001yn1yn3



15 ₁₀ 2 2 ₁₁₆₆₄₀

 Y

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2) Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabola which are presented in the Table: 3 below:

Table: 3 Parabolas

S. No Parabola



_X,Y



1 _{27 10} 2 ₂₉₁₆



 Y

X



79x2n2x2n354,xn225xn1



2 _{27 10} 2 ₂₉₁₆



 Y

X



3001x2n379x2n454,25xn3949xn2



3 _{1026 10} 2 _4210704



 Y

X



3001x2n2x2n42052,xn3949xn1



4 _{9 10} 2 ₃₂₄



 Y

X



20x2n2y2n218,yn12xn1



5 _{171 10} 2 ₁₁₆₉₆₄



 Y

X



500x2n2y2n3342,yn2158xn1



6 _{6489 10} 2 _168428484



 Y

X



18980x2n2y2n412978,yn36002xn1



7 _{171 10} 2 ₁₁₆₉₆₄



 Y

X



20x2n379y2n2342,25yn12xn2



8 _{9 10} 2 ₃₂₄



 Y

X



500x2n379y2n318,25yn2158xn2



9 _{171 10} 2 ₁₁₆₉₆₄



 Y

X



18980x2n379y2n4342,25yn36002xn2



10 _{6489 10} 2 _168428484



 Y

X



20x2n43001y2n212978,949yn12xn3



11 _{171 10} 2 ₁₁₆₉₆₄



 Y

X



500x2n43001y2n3342,949yn2158xn3



12 _{9 10} 2 ₃₂₄



 Y

X



18980x2n43001y2n418,949yn36002xn3



13 ₅₄₀ 2 ₁₁₆₆₄₀

 Y

X



y2n325y2n2108,79yn1yn2



14 ₂₀₅₂₀ 2 _168428160

 Y

X



y2n4949y2n24104,3001yn1yn3



15 ₅₄₀ 2 ₁₁₆₆₄₀

 Y

X



25y2n4949y2n3108,3001yn279yn3



3) Relations between the solutions and special figurate numbers

a)











3, 3, 1



2

2 1 , 3 5 2

, 3

3 _{40 36}

9 pyt x  pxt y_  t xt y_

b)











3, 3, 1



2

2

, 3 3 2

1 , 3 5

36

360 _

    

 x x y y x

y t p t t t

p

c)





2



5 _{3,2 2}



2



_{3, 3,2 2}



2

, 3 4

1 40 36

36 p_y t _x  p_xt _y  t _xt _y

d)











3, 3,2 2



2

2

, 3 4

1 2

2 2 , 3 5

36

1440 _{ }

    

 x x y y x

y t p t t t

p

e)











3, 1 3,2 2



2

2

2 2 , 3 3 2

1 , 3 4

1  10   

  x  x y  x  y

y t p t t t

46

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f)











3, 1 3,2 2



2

2 1 , 3 4

1 2

2 2 , 3 3

4

160 _{   }

    

 x x y y x

y t p t t t

p

g) Let {ns1}and{ms1}be sequences of positive integers defined by



1



2 1 }

{ns₁  xs₁ , 1



1



2 1 }

{ms_  ys_

Note that,

6

480t3,ns₁ is a nasty number

2 160t3,n_s1t6,m_s1ms1

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