Observation on the Positive Pell Equation

Observation on the Positive Pell Equation

10

15

2

2





x

y

M. A. Gopalan1, A. Sathya2, A. Nandhinidevi3*

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 Diophantine equation represented by the positive Pellian _y2 _15_x2_10_{is analyzed for its}

non-zero 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 binary quadratic equation of the form y2  Dx2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 [5-16]. In this communication, yet another an interesting equation given by y2 15x210 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

2 15 2 10



 x

y (1)

The smallest positive integer solutions of (1) are

x01,y05

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

15 1

2 2



 x

y (2) whose initial solution is given by

1 ~

0 

x ,~y0 4

The general solution



x~n,y~n



of (2) is given by

n

n g

x

15 2

1

~ _{ ,}

n

n f

y

2 1

~ _

where

1 1

) 15 4 ( ) 15 4

(     

 n n

n

f

(4 15) (4 15) , 1,0,1,...

1 1

  

 

   n

g_n n n

Applying Brahmagupta lemma between



x0,y0



and



~xn,~yn



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

n n

n f g

x

15 2

5 2

1

1  



n n

n f g

y

2 15 2

5

1  



The recurrence relations satisfied by the solutions xand yare given by xn38xn2xn10

[image:2.612.198.416.103.211.2]

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

Table 1: Numerical Examples

n

n

x

yn

0 9 35

1 71 275

2 559 2165

3 4401 17045

4 34649 134195

From the above table, we observe some interesting relations among the solutions which are presented below: Both xn and yn values are odd.

A. Relations Among The Solutions Are Given Below 1) xn3 8xn2  xn1  0

2) yn3 31xn2 4xn1 0

3) 8yn131xn1xn3 0

4) 8yn3 31xn3  xn1 0

5) xn2 4xn1 yn1 0

6) yn2 4yn115xn1 0

7) yn3 31yn1120xn1 0

8) 4xn2  xn1 yn2  0

9) xn3  xn1 2yn2  0

10) 4yn3 31yn215xn1 0

11) yn1 31xn2 4xn3  0

12) yn2  4xn2  xn3  0

13) yn3  xn2 4xn3  0

14) 4yn2  yn115xn2 0

15) yn3yn130xn3 0

16) yn3 4yn2 15xn2  0

17) yn3 4xn131xn2 0

18) 4xn3  xn2  yn3  0

19) 31yn2 4yn1 15xn30

20) 31yn3  yn1120xn3 0

21) 4yn3  yn2 15xn3  0

22) yn3 8yn2  yn1  0 23) 445yn3976yn231307yn10

24) 14640xn313543yn3952753yn10

25) 14640xn2 1717yn3 120787 yn10

B. Each Of The Following Expressions Represents A Nasty Number 1)



12 6x2n3 42x2n2



2)



96 6 2 4 330 2 2



8 1

  

 x n x n

3)



12 6y2n2 18x2n2



4)



48 6 2 3 162 2 2



4 1

  

 y n x n

5)



372 6 2 4 1278 2 2



31 1

  

 y n x n

6)



1242x2n4330x2n3



7)



48 42 2 2 18 2 3



4 1

  

 y n x n

8)



12 42y2n3 162x2n3



9)



48 42 2 4 1278 2 3



4 1

  

 y n x n

10)



372 330 2 2 18 2 4



31 1

  

 y _n x _n

11)



48 330 2 3 162 2 4



4 1

  

 y _n x _n

12)



12330y2n41278x2n4



13)



60 54 2 2 6 2 3



5 1

  

 y _n y _n

14)



480 426 2 2 6 2 4



40 1

  

 y n y n

15)



60 426 2 3 54 2 4



5 1

  

 y n y n

III.EACHOFTHEFOLLOWINGEXPRESSIONSREPRESENTSACUBICALINTEGER

1)



x3n4 7x3n3 3xn2 21xn1



2)



3 5 55 3 3 3 3 165 1



8 1

 



  n  n  n

n x x x

x

3)



y3n3 3x3n3 9xn1 3yn1



4)



3 4 27 3 3 81 1 3 2



4 1

  

  n  n  n

n x x y

y

5)



3 5 213 3 3 3 3 639 1



31 1

 



  n  n  n

n x y x

y

6)



7x3n555x3n4 21xn3165xn2



7)



7 3 3 3 3 4 21 1 9 2



4 1

  

  n  n  n

n x y x

y

8)



7y3n427x3n4 81xn221yn2



9)



7 3 5 213 3 4 639 2 21 3



4 1

 



  n  n  n

n y x y

y

10)



55 3 3 3 3 5 9 3 165 1



31 1

 



  n  n  n

n x x y

y

11)



55 3 4 27 3 5 81 3 165 2



4 1

 



  n  n  n

n x x y

y

13)



9 3 3 3 4 27 1 3 2



5 1   

  n  n  n

n y y y

y

14)



71 3 3 3 5 213 1 3 3



40 1

  

  n  n  n

n y y y

y

15)



71 3 4 9 3 5 213 2 27 3



5 1

 



  n  n  n

n y y y

y

IV.EACHOFTHEFOLLOWINGEXPRESSIONSREPRESENTSABIQUADRATICINTEGER

1)



x4n57x4n428x2n2 4x2n36



2)



55 220 4 48



8 1 4 2 2 2 4 4 6

4n  xn  x n  x n 

x

3)



y4n43x4n412x2n24y2n26



4)



27 4 108 24



4 1 2 2 3 2 4 4 5

4n  xn  y n  xn 

y

5)



213 4 852 186



31 1 2 2 4 2 4 4 6

4n  xn  yn  xn 

y

6)



7x4n6220x2n355x4n528x2n46



7)



7 3 12 28 24



4 1 2 2 3 2 5 4 4

4n  x n  x n  y n 

y

8)



7y_4n528y_2n327x_4n5108x_2n36



9)



7 28 213 852 24



4 1 3 2 5 4 4 2 6

4n  y n  xn  xn 

y

10)



55 3 12 220 186



31 1 2 2 4 2 6 4 4

4n  x n  xn  yn 

y

11)



220 55 27 108 24



4 1 4 2 6 2 5 4 3

2n  yn  xn  xn 

y

12)



55y_4n6220y_2n4213x_4n6852x_2n46



13)



9 36 4 30



5 1 3 2 2 2 5 4 4

4n y n  y n  y n 

y

14)



71 284 4 240



40 1 4 2 2 2 6 4 4

4n y n  yn  yn 

y

15)



71 9 284 36 30



5 1 4 2 3 2 6 4 5

4n  xn  yn  yn 

y

V. EACHOFTHEFOLLOWINGEXPRESSIONSREPRESENTSAQUINTICINTEGER

16)



x5n67x5n5 5x3n5 35x3n3 70xn110xn2



17)



5 7 55 5 5 5 3 5 275 3 3 550 1 10 3



8 1     

  n  n  n  n  n

n x x x x x

x

18)



y5n53x5n5 5y3n315x3n330xn110yn1



19)



5 6 27 5 5 5 3 4 135 3 3 270 1 10 2



4 1     

  n  n  n  n  n

n x x x x y

y

20)



5 7 213 5 5 1065 3 5 5 3 5 2130 1 10 3



31 1     

  n  n  n  n  n

n x x y x y

y

21)



7y5n755x5n635x3n5275x3n4550xn2 70xn3



22)



7 5 5 3 5 6 15 3 4 35 3 4 30 2 70 1



4 1     

  n  n  n  n  n

n x x x x y

y

24)



7 5 7 213 5 6 10653 4 35 3 5 2130 2 70 3



4 1

  

 

  n  n  n  n  n

n x x y x y

y

25)



5 6 27 5 5 5 3 4 135 3 3 270 1 10 2



4 1

  

 

  n  n  n  n  n

n x x x x y

y

26)



55 5 5 3 5 7 15 3 5 275 3 3 30 3 550 1



31 1

 

 



  n  n  n  n  n

n x x y x y

y

27)



55 5 6 27 5 7 135 3 5 275 3 4 270 3 550 2



4 1

 

 



  n  n  n  n  n

n x x y x y

y

28)



55y5n7213x5n71065x3n5275y3n52130xn3550yn3



29)



9 5 5 5 6 5 3 4 45 3 3 90 1 10 2



5 1

 

 



  n  n  n  n  n

n y y y y y

y

30)



71 5 5 5 7 355 3 3 5 3 5 710 1 10 3



40 1

 

 



  n  n  n  n  n

n y y y y y

y

31)



71 5 6 9 5 7 355 3 4 45 3 5 710 2 90 3



5 1

 

 



  n  n  n  n  n

n y y y y y

y

VI. REMARKABLEOBSERVATIONS

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

[image:5.612.106.511.339.715.2]

which are presented in Table 2 below:

Table 2: Hyperbolas

S.NO Hyperbola (X,Y)

1 ₅ 2 ₃ 2 ₂₀



 Y

X



xn2 7xn1 , 9xn1 xn2



2 ₅ 2 ₃ 2 ₁₂₈₀



 Y

X



xn355xn1 , 71xn1xn3



3 ₅ 2 ₃ 2 ₂₀



 Y

X



yn13xn1 , 5xn1  yn1



4 ₅ 2 ₃ 2 ₃₂₀



 Y

X



yn2 27xn1 , 35xn1 yn2



5 _5X2_3Y2 _19220





3 1 1

3 213  , 275  

  n n  n

n x x y

y

6 ₅ 2 ₃ 2 ₂₀



 Y

X



7xn355xn2 , 71xn29xn3



7 ₅ 2 ₃ 2 ₃₂₀



 Y

X



7

y

_n1



3

x

_n2

,

5

x

_n2



9

y

_n1



8 ₅ 2 ₃ 2 ₂₀



 Y

X



7yn227xn2 ,35xn29yn2



9 ₅ 2 ₃ 2 ₃₂₀



 Y

X



7yn3213xn2,275xn29yn3



10 ₅ 2_3 2 _19220

Y

X



55yn13xn3 ,5xn371yn1



11 ₅ 2 ₃ 2 ₃₂₀



 Y

X



55yn227xn3,35xn371yn2



12 ₅ 2 ₃ 2 ₂₀



 Y

X



55yn3213xn3,275xn371yn3



13 ₁₁₁₆₃ 2 ₂₀ 2 _71443200



 Y

X



71yn1yn3 ,55yn1yn3



14 ₃ 2 ₅ 2 ₃₀₀



 Y

X



9yn1 yn2 , yn27yn1



15 ₃ 2 ₅ 2 ₃₀₀



 Y

[image:6.612.106.508.115.434.2]

2) 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: Parabolas

S.NO Parabola (X,Y)

1 ₃ 2 _{5 10}



 X

Y



x2n3 7x2n2 , 9xn1 xn3



2 ₃ 2 _{40 640}



 X

Y



x2n4 55x2n2 , 71xn1xn3



3 ₃ 2 _{5 10}



 X

Y



y2n2 3x2n2 , 5xn1 yn1



4 ₃ 2 _{20 160}



 X

Y



y2n3 27x2n2 ,35xn1 yn2



5 ₃ 2 _{155 9610}



 X

Y



y2n4213x2n2 ,275xn1yn3



6 ₃ 2 _{5 10}



 X

Y



7x2n455x2n3,71xn29xn3



7 ₃ 2 _{20 160}



 X

Y



7y2n23x2n3 , 5xn2 9yn1



8 ₃ 2 _{5 10}



 X

Y



7y2n327x2n3,35xn29yn2



9 ₃ 2 _{20 160}



 X

Y



7y2n4213x2n3,275xn29yn3



10 ₃ 2 _{155 9610}



 X

Y



55y2n23x2n4 ,5xn371yn1



11 ₃ 2 _{20 160}



 X

Y



55y2n327x2n4,35xn371yn2



12 ₃ 2 _{5 10}



 X

Y



55y2n4213x2n4,275xn371yn3



13 ₃ 2 _{5 10}



 X

Y



9

y

_2n2



y

_2n3

,

y

_n2



7

y

_n1



14 2 _{22326 1786080}



 X

Y



71y2n2y2n4 ,55yn1yn3



15 2 _{3 30}



 X

Y



71y2n39y2n4 ,7yn355yn2



VII. CONCLUSION

In this paper, we have presented infinitely many integer solutions for the positive Pell equation y215x210 . As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of Pell equations and determine their integer solutions along with suitable properties.

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

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