On the Positive Pell Equation

On the Positive Pell Equation

y

2



32

x

2



41

K. Meena1, S. Vidhyalakshmi2, K. Radha 3 1

Former VC, Bharathidasan University, Trichy-620 024.Tamilnadu, India 2

Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, India. 3

M.Phil Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, India

Abstract: The binary quadratic equation represented by the positive Pellian

y

2



32

x

2



41

is analyzed for its distinct integer

solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbola and parabola.

Keywords: Binary quadratic, hyperbola, integral solutions, parabola, Pell equation. 2010 mathematics subject classification: 11D09

I. INTRODUCTION

A binary quadratic equation of the form

y

2



Dx

2



1

where D is non-square positive integer has been studied by various mathematicians for its non-trivial integral solutions when D takes different integral values [1-2]. For an extensive review of various

problems, one may refer [3-12]. In this communication, yet another interesting hyperbola given by

y

2



32

x

2



41

is considered

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

A. Method of Analysis

The positive Pell equation representing hyperbola under consideration is

41

32

2

2





x

y

(1)

whose smallest positive integer solution is

13

,

2

₀

0



y



x

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

1

32

2

2





x

y

whose general solution is given by

n n n

n

g

y

f

x

2

1

~

,

2

8

1

~

_

_

where





1





1

2

12

17

2

12

17













n n

n

f



17



12

2



1





17



12

2



1

,





1

,

0

,

1

,

2

...



 

n

g

n n n

Applying Brahmagupta lemma between



x

₀

,

y

₀



and



x

~

_n

,

~

y

_n



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

n n

n

f

g

x

2

8

13

1







n n

n

f

g

y

8

2

2

13

1







The recurrence relations satisfied by x and y are given by

0

34

2 1

3











 n n

n

x

x

x

0

34

_{2 1}

3











 n n

n

y

y

y

Table: 1 Numerical examples

n

1



n

x

y

_n1

-1 2 13

0 73 413

1 2480 14029

2 84247 476573

3 2861918 16189453

4 97220965 549964829

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

1)

x

_n1 is alternatively even and odd,

y

_n1is always odd.

2) Relations among the solutions

a)

x

_n1



17

x

_n2



3

y

_n2



0

b)

17

x

_n1



577

x

_n2



3

y

_n3



0

c)

x

_n1



x

_n3



6

y

_n2



0

d)

x

_n1



577

x

_n3



102

y

_n3



0

e)

17

x

_n1



x

_n2



3

y

_n1



0

f)

577

x

n1



x

n3



102

y

n1



0

g)

96

x

_n1



17

y

_n1



y

_n2



0

h)

3264

x

_n1



577

y

_n1



y

_n3



0

i) 96

x

_n1



577

y

_n2



17

y

_n3



0

j)

1154

x

_n2



17

x

_n3



3

y

_n3



0

k)

17

x

n2



x

n3



3

y

n2



0

l)

x

_n2



17

x

_n3



3

y

_n3



0

m)

96

x

n2



17

y

n2



y

n1



0

n)

192

x

_n2



y

_n1



y

_n3



0

o)

96

x

_n2



y

_n3



17

y

_n2



0

p) 192

x

n2



y

n3



697

y

n1



0

q)

96

x

_n3



577

y

_n2



17

y

_n1



0

r)

3264

x

n3



577

y

n3



y

n1



0

s)

96

x

_n3



y

_n2



17

y

_n3



0

3) Each of the following expressions represents a nasty number

a)



13

413

123



41

4

2 2 3

2n



x

n



x

b)



13

14029

4182



697

2

2 2 4

2n



x

n



x

c)



39

192

123



41

4

2 2 2

2n



x

n



y

d)



26

4672

1394



697

6

2 2 3

2n



x

n



y

e)



39

238080

70971



23657

4

2 2 4

2n



x

n



y

f)



413

14029

123



41

4

3 2 4

2n



x

n



x

g)



1239

192

2091



697

4

3 2 2

2n



x

n



y

h)



1239

7008

123



41

4

3 2 3

2n



x

n



y

i)



1239

238080

2091



697

4

3 2 4

2n



x

n



y

j)



42087

192

70971



23657

4

4 2 2

2n



x

n



y

k)



42087

7008

2091



697

4

4 2 3

2n



x

n



y

l)



42087

238080

123



41

4

4 2 4

2n



x

n



y

m)



73

2

123



41

4

3 2 2

2n



y

n



y

n)



1240

2091



697

4

4 2 2

2n



y

n



y

o)



2480

73

123



41

4

4 2 3

2n



y

n



y

4) Each of the following expressions represents a cubical integer

a)



78

₂

2478

₁

26

_{3 4}

826

_{3 3}



123

1

 







n



n



n

n

x

x

x

x

b)



39

₃

42087

₁

13

_{3 5}

14029

_{3 3}



2091

1

 







n



n



n

n

x

x

x

x

c)



78

₁

384

₁

26

_{3 3}

128

_{3 3}



41

1

 







n



n



n

n

x

y

x

y

d)



78

₂

14016

₁

26

_{3 4}

4672

_{3 3}



697

1

 







n



n



n

n

x

y

x

e)



78

₃

476160

₁

26

_{3 5}

158720

_{3 3}



23657

1

  





n



n



n

n

x

y

x

y

f)



2478

₃

84174

₂

826

_{3 5}

28058

_{3 4}



123

1

 







n



n



n

n

x

x

x

x

g)



2478

₁

384

₂

826

_{3 3}

128

_{3 4}



697

1

 







n



n



n

n

x

y

x

y

h)



2478

₂

14016

₂

826

_{3 4}

4672

_{3 4}



41

1

 







n



n



n

n

x

y

x

y

i)



2478

₃

476160

₂

826

_{3 5}

158720

_{3 4}



697

1

 







n



n



n

n

x

y

x

y

j)



84174

₁

384

₃

28058

_{3 3}

128

_{3 5}



23657

1

 







n



n



n

n

x

y

x

y

k)



84174

2

14016

3

28058

3 4

4672

3 5



697

1

 







n



n



n

n

x

y

x

y

l)



84174

₃

476160

₃

28058

_{3 5}

158720

_{3 5}



41

1

 







n



n



n

n

x

y

x

y

m)



438

₁

12

₂

146

_{3 3}

4

_{3 4}



123

1

 







n



n



n

n

y

y

y

y

n)



7440

₁

6

₃

2480

_{3 3}

2

_{3 5}



2091

1

 







n



n



n

n

y

y

y

y

o)



14880

₂

438

₃

4960

_{3 4}

146

_{3 5}



123

1

 







n



n



n

n

y

y

y

y

5) Each of the following expressions represents a bi-quadratic integer

a)



104

3304

26

826

738



123

1

4 4 5 4 2 2 3

2n



x

n



x

n



x

n



x

b)



52

56116

13

14029

12546



2091

1

4 4 6 4 2 2 4

2n



x

n



x

n



x

n



x

c)



104

512

26

128

246



41

1

4 4 4 4 2 2 2

2n



x

n



y

n



x

n



y

d)



104

18688

26

4672

4182



697

1

4 4 5 4 2 2 3

2n



x

n



y

n



x

n



y

e)



104

634880

26

158720

141942



23657

1

4 4 6 4 2 2 4

2n



x

n



y

n



x

n



y

f)



3304

112232

826

28058

738



123

1

5 4 6 4 3 2 4

2n



x

n



x

n



x

n



x

g)



3304

512

826

128

4182



697

1

5 4 4 4 3 2 2

2n



x

n



y

n



x

n



y

h)



3304

18688

826

4672

246



41

1

5 4 5 4 3 2 3

2n



x

n



y

n



x

n



i)



3304

634880

826

158720

4182



697

1

5 4 6 4 3 2 4

2n



x

n



y

n



x

n



y

j)



112232

512

28058

128

141942



23657

1

6 4 4 4 4 2 2

2n



x

n



y

n



x

n



y

k)



112232

18688

28058

4672

4182



697

1

6 4 5 4 4 2 3

2n



x

n



y

n



x

n



y

l)



112232

634880

28058

158720

246



41

1

6 4 6 4 4 2 4

2n



x

n



y

n



x

n



y

m)



584

16

146

4

738



123

1

5 4 4 4 3 2 2

2n



y

n



y

n



y

n



y

n)



9920

8

2480

2

12546



2091

1

6 4 4 4 4 2 2

2n



y

n



y

n



y

n



y

o)



19840

584

4960

146

738



123

1

6 4 5 4 4 2 3

2n



y

n



y

n



y

n



y

6) Each of the following expressions represents a quintic integer

a)



260

2

8260

1

130

3 4

4130

3 3

26

5 6

826

5 5



123

1

    





n



n



n



n



n

n

x

x

x

x

x

x

b)



130

3

140290

1

65

3 5

70145

3 3

13

5 7

14029

5 5



2091

1

    





n



n



n



n



n

n

x

x

x

x

x

x

c)



260

₁

1280

₁

130

_{3 3}

640

_{3 3}

26

_{5 5}

128

_{5 5}



41

1

    





n



n



n



n



n

n

x

y

x

y

x

y

d)



260

₂

46720

₁

130

_{3 4}

23360

_{3 3}

26

_{5 6}

4672

_{5 5}



697

1

    





n



n



n



n



n

n

x

y

x

y

x

y

e)



260

₃

1587200

₁

130

_{3 5}

793600

_{3 3}

26

_{5 7}

158720

_{5 5}



23657

1

    





n



n



n



n



n

n

x

y

x

y

x

y

f)



8260

₃

280580

₂

4130

_{3 5}

140290

_{3 4}

826

_{5 7}

28058

_{5 6}



123

1

    





n



n



n



n



n

n

x

x

x

x

x

x

g)



12390

₁

1920

₂

128

_{5 6}

826

_{5 5}



697

1

 







n



n



n

n

x

x

y

y

h)



8260

₂

46720

₂

4130

_{3 4}

23360

_{3 4}

826

_{5 6}

4672

_{5 6}



41

1

    





n



n



n



n



n

n

x

y

x

y

x

y

i)



8260

₃

1587200

₂

4130

_{3 5}

793600

_{3 4}

826

_{5 7}

158720

_{5 6}



697

1

    





n



n



n



n



n

n

x

y

x

y

x

y

j)



280580

₁

1280

₃

140290

_{3 3}

640

_{3 5}

28058

_{5 5}

128

_{5 7}



23657

1

    





n



n



n



n



n

n

x

y

x

y

x

y

k)



280580

2

46720

3

23360

3 5

140290

3 4

28058

5 6

4672

5 7



697

1

    





n



n



n



n



n

n

x

x

y

y

x

y

l)



280580

3

1587200

3

140290

3 5

793600

3 5

28058

5 7

158720

5 7



41

1

    





n



n



n



n



n

n

x

y

x

y

x

m)



1460

₁

40

₂

730

_{3 3}

20

_{3 4}

146

_{5 5}

4

_{5 6}



123

1

 

 







n



n



n



n



n

n

y

y

y

y

y

y

n)



24800

₁

20

₃

12400

_{3 3}

10

_{3 5}

2480

_{5 5}

2

_{5 7}



2091

1

 

 







n



n



n



n



n

n

y

y

y

y

y

y

o)



49600

₂

1460

₃

24800

_{3 4}

730

_{3 5}

4960

_{5 6}

146

_{5 7}



123

1

 

 







n



n



n



n



n

n

y

y

y

y

y

y

B. 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 Hyperbolas

_

_X

_,

_Y

_

1 2

₃₂

2

₆₀₅₁₆





Y

X



26

x

n2



826

x

n1

,

146

x

n1



4

x

n2



2 2

₃₂

2

_69956496





Y

X



26

x

n3



28058

x

n1

,

4960

x

n1



4

x

n3



3 2

₃₂

2

₆₇₂₄





Y

X



26

y

n1



128

x

n1

,

26

x

n1



4

y

n1



4 2

₃₂

2

_1943236





Y

X



26

y

n2



4672

x

n1

,

826

x

n1



4

y

n1



5 2

₃₂

2

_2238614596





Y

X

(

26

y

_n3



158720

x

_n1

,

28058

x

_n1



4

y

_n3)

6 2

₃₂

2

₆₀₅₁₆





Y

X



826

x

n3



28058

x

n2

,

4960

x

n2



146

x

n3



7 2

₃₂

2

_1943236





Y

X



826

y

n1



128

x

n2

,

26

x

n1



146

y

n1



8 2

₃₂

2

₆₇₂₄





Y

X



826

y

n2



4672

x

n2

,

826

x

n2



146

y

n2



9 2

₃₂

2

_1943236





Y

X



826

y

n3



158720

x

n2

,

28058

x

n2



146

y

n3



10 2

₃₂

2

_2238614596





Y

X



28058

y

n1



128

x

n3

,

26

x

n3



4960

y

n1



11

32

1943236

2 2





Y

X



28058

y

n2



4672

x

n3

,

826

x

n3



4960

y

n2



12 2

₃₂

2

₆₇₂₄





Y

X



28058

y

n3



158720

x

n3

,

28058

x

n3



4960

y

n3



13

₃₂

2 2

_1936512





Y

X



146

y

n1



4

y

n2

,

26

y

n2



826

y

n1



14

₃₂

2 2

_2238607872





Y

X



4960

y

n1



4

y

n3

,

26

y

n3



28058

y

n1



15

₃₂

2 2

_1936512





Y

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

Parabolas

_

_X

_,

_Y

_

1

₁₂₃

₃₂

2

₃₀₂₅₈





Y

X



26

x

2n3



826

x

2n2

,

146

x

n1



4

x

n2



2

₄₁₈₂

₃₂

2

_34978248





Y

X



26

x

2n4



28058

x

2n2

,

4960

x

n1



4

x

n3



3

₄₁

₃₂

2

₃₃₆₂





Y

X



26

y

2n2



128

x

2n2

,

26

x

n1



4

y

n1



4

₆₉₇

₃₂

2

₉₇₁₆₁₈





Y

X



26

y

2n3



4672

x

2n2

,

826

x

n1



4

y

n2



5

₂₃₆₅₇

₃₂

2

_1119307298





Y

X



26

y

2n4



158720

x

2n2

,

28058

x

n1



4

y

n3



6

₁₂₃

₃₂

2

₃₀₂₅₈





Y

X



826

x

2n4



28058

x

2n3

,

4960

x

n2



146

x

n3



7

₆₉₇

₃₂

2

₉₇₁₆₁₈





Y

X



826

y

2n2



128

x

2n3

,

26

x

n2



146

y

n1



8

₄₁

₃₂

2

₃₃₆₂





Y

X



826

y

2n3



4672

x

2n3

,

826

x

n2



146

y

n2



9

₆₉₇

₃₂

2

₉₇₁₆₁₈





Y

X



826

y

2n4



158720

x

2n3

,

28058

x

n2



146

y

n3



10

₂₃₆₅₇

₃₂

2

_1119307298





Y

X



28058

y

2n2



128

x

2n4

,

26

x

n3



4960

y

n1



11

₆₉₇

₃₂

2

₉₇₁₆₁₈





Y

X



28058

y

_2n3



4672

x

_2n4

,

826

x

_n3



4960

y

_n2



12

₄₁

₃₂

2

₃₃₆₂





Y

X



28058

y

2n4



158720

x

2n4

,

28058

x

n3



4960

y

n3



13

₃₉₃₆

2

₉₆₈₂₅₆





Y

X



146

y

2n2



4

y

2n3

,

26

y

n2



826

y

n1



14

₁₃₃₈₂₄

2

_1119303936





Y

X



4960

y

2n2



4

y

2n4

,

26

y

n3



28058

y

n1



15

₃₉₃₆

2

₉₆₈₂₅₆





Y

REFERENCES

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

[2] David M.Burton, Elementary Number Theory, Tata Mc Graw Hill Publishing Company Limited, New Delhi-2002.

[3] Geetha T., Gopalan M.A., Vidhyalakshmi S., Observation on the hyperbola,

y

2



72

x

2



1

, Scholars Journal of Physics, Mathematics and Statistics, 1(1), 2014, 1-13.

[4] Gopalan M.A, Vidhyalakshmi S., Umarani J., Remarkable Observations on the hyperbola

y

2



24

x

2



1

, Bulletin of Mathematics and Statistics Research, 1, 2014, 9-12.

[5] Gopalan M.A , Vidhyalakshmi S, Premalatha E, Vanitha M, Observation on the hyperbola,

y

2



220

x

2



9

, Star Research Journal, Vol.4, Issue3(2), 2016, 12-16.

[6] Ramya S, Kavitha A. On the positive pell equation,

y

2



90

x

2



31

, Journal of Mathematics and informatics, 11(1), 2017, 11-17.

[7] Sumathi G., Observations on the hyperbola,

y

2



150

x

2



16

, International Journal of Recent Trends in Engineering and Research, 3(9), 2017, 198-206.

[8] Meena K, Gopalan M.A., Sivaranjani V, On the positive pell equation

y

2



102

x

2



33

. International Journal of Advanced Education and Research 2(1), 2017, 91-96

[9] Gopalan M.A, Devibala S., Swetha T., On the positive pell equation

y

2



5

x

2



4

, IOSR Journal of Mathematics, Vol 13, Issue 5, Ver.1, 2017, 65-69.

[10] Usha Rani T.R, Dhivya K, On the positive pell equation

y

2



14

x

2



18

, International Journal of Academic Research and Development, Volume 3, Issue 3, 2018, 43-50.

[11] Shreemathi Adiga, Anusheela N. and Gopalan M.A., On the Hyperbola

y

2



3

x

2



4

, IJLEMR, 03(06), 2018, 45-48