On the Negative Pellian Equation $y^2=110 x^2-29$

Vol. 11, 2017, 63-71

ISSN: 2349-0632 (P), 2349-0640 (online) Published 11 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/jmi.v11a9

63

Journal of

On the Negative Pellian Equation y

2

=110 x

2

-29

R.Suganya1 and D.Maheswari2

Department of Mathematics, Shrimati Indira Gandhi College Trichy-620002, Tamilnadu, India.

1

e-mail: suganyaccc@yahoo.in; 2e-mail: matmahes@gmail.com

Received 1 November 2017; accepted 8 December 2017

Abstract. The hyperbola represented by the binary quadratic equation y2=110 x2-29 is analyzed for finding its non-zero distinct integer solutions. A few interesting relations among its solutions are presented. Also knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special pythagorean triangle is constructed.

Keywords:binary quadratic, hyperbola, negative pell equation, integral solution. AMS Mathematics Subject Classification (2010): 11D09

1. Introduction

Diophantine equation of the form = + 1, 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 all the world, since antiquity, J.L Lagrange proved that all positive pell equation = + 1 has infinitely many distinct integer solutions where as the negative pell equation = − 1does not always have a solution. In [1], an elementary proof of a ceriterium for the solvability of the pell equation − = −1 where D is any positive non-square integer has been presented. For examples the

equationsy2 =3x2−1,y2 =7x2 −4 have no integer solutions, whereas

1 65 2

2 = −

x

y ,y2 =202x2 −1 have integer solutions. In this context, one may refer [2-12]. More specifically, one may refer “The online encyclopedia of Integer sequences” (A031396, A130226, A031398) for values of D for which the negative pell equation

1 2 2 = −

Dx

y is solvable or not. In this communication, the negative pell equation given by y2=110x2-29 is considered and infinitely many integer solutions are obtained. A few interesting relations among the solution are presented. Also knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also employing the solutions of the given equation, special pythagorean triangle is constructed.

64

The negative Pell equation representing hyperbola under consideration is

110 29 2 2 = _x −

y ₍₁₎

whose smallest positive integer solution is x₀ =1,y₀ =9

To obtain, the other solutions of (1), consider the pell equation y2 =110x2 +1

whose general solution is given by

~ _{2 110},~ ₂

n n n n

f y g

x = =

where

(

21 2 110

) (

21 2 110

)

,

1

1 +

+

− + +

= n n

n f

(

21 2 110

) (

21 2 110

)

,

1

1 +

+ _{− −}

−

= n n

n

g _{n = -1, 0, 1, 2……}

Applying Brahmagupta lemma between the solutions of (x0,y0) and (x~n,~yn), the other

integer solutions of (1) are given by

x_n f_n g_n 110 2

9 2

1 1 = +

+ (2)

yn fn gn

110 2

110 2

9 1 = +

+ (3)

Thus (2) and (3) represent the non-zero distinct integer solutions of (1)

The recurrence relations satisfied by the values of xn+1 and yn+1 are respectively.

0 42

0 42

1 2 3

1 2 3

= + −

= + −

+ + +

+ + +

n n n

n n n

y y y

x x x

A few numerical examples are given in the following table 1:

Table 1: Examples n

1 +

n

x y_n+1 -1 1 9 0 39 409 1 1637 17169 2 68715 720689

2.1. A few interesting relations among the solutions are given below 2y_n+2 −21x_n+2 +x_n+1 =0

638y_n+3 −281039x_n+2 +6699x_n+1 =0 42x_n+2−x_n+1−x_n+3 =0

On the Negative Pellian Equation y =110 x-29

65 4y_n+2−x_n+3+x_n+1=0

84y_n+3 −881x_n+3 +x_n+1 =0 x_n+2 −21x_n+1−2y_n+1 =0 y_n+2−220x_n+1−21y_n+1 =0 y_n+3−9240x_n+1−881y_n+1=0

66990x_n+2−18236819x_n+1−38201y_n+2 =0 66990x_n+3−765478179x_n+1−1603461y_n+2=0 33495y_n+1−2101055x_n+1+4400y_n+2 =0

66990y_n+3−8028402470x_n+1+16817240y_n+2 =0

281039y_n+2 −6699y_n+3 +70180x_n+1 =0 6380x_n+1 −1888590x_n+2+44990x_n+3 =0 2yn+2 −xn+3 +21xn+2 =0

6699x_n+3−281039x_n+2−638y_n+1=0 21yn+2 −220xn+2 −yn+1 =0

y_n+1−21y_n+2 −2y_n+2 =0 y_n+3 −220x_n+2 −21y_n+2 =0

6699x_n+1−281039x_n+2+638y_n+3 =0 13398y_n+1−562078y_n+2+140360x_n+3=0 y_n+1−881y_n+3+9240x_n+3 =0

y_n+2−21y_n+3 +220x_n+3 =0

70180x_n+3−281039y_n+2 +6699y_n+1 =0 140360x_n+1−13398y_n+3+562078y_n+2 =0 y_n+1−42y_n+2+y_n+3 =0

2.2. Each of the following expressions represents a Nasty number

[

116 818 2 2 18 2 3

]

58

6

+ + −

+ x n x n

[

4872 34338 2 2 18 2 4

]

2436

6

+ + −

+ x n x n

[

58 220 2 2 18 2 2

]

29

6

+ + −

+ x n y n

[

1218 8580 2 2 18 2 3

]

609

6

+ + −

66

[

51098 360140 2 2 18 2 4

]

25549

6

+ + −

+ x n y n

[

116 34338 2 3 818 2 4

]

58

6

+ + −

+ x n x n

[

1218 220 2 3 818 2 2

]

609 6

+ + −

+ x n y n

[

58 8580 2 3 818 2 3

]

29

6

+ + −

+ x n y n

[

1218 360140 2 3 818 2 4

]

609

6

+ + −

+ x n y n

[

51098 220 2 4 34338 2 2

]

25549

6

+ + −

+ x n y n

[

1218 8580 2 4 34338 2 3

]

609

6

+ + −

+ x n y n

[

58 360140 2 4 34338 2 4

]

29

6

+ + −

+ x n y n

[

12760 220 2 3 8580 2 2

]

6380

6

+ + −

+ y n y n

[

535920 220 2 4 360140 2 2

]

267960

6

+ + −

+ y n y n

[

12760 8580 2 4 360140 2 3

]

6380

6

+ + −

+ y n y n

2.3. Each of the following expressions represents a cubical integer

[

(

818 3 3 18 3 4

) (

3818 1 18 2

)

]

58

1

+ +

+

+ − n + n − n

n x x x

x

[

(

34338 3 3 18 3 5

) (

334338 1 18 3

)

]

2436

1

+ +

+

+ − n + n − n

n x x x

x

[

(

220 3 3 18 3 3

) (

3220 1 18 1

)

]

29

1

+ +

+

+ − n + n − n

n y x y

x

[

(

8580 3 3 18 3 4

) (

38580 1 18 2

)

]

609

1

+ +

+

+ − n + n − n

n y x y

x

[

(

360140 3 3 18 3 5

) (

3360140 1 18 3

)

]

25549

1

+ +

+

+ − n + n − n

n y x y

x

[

(

34338 3 4 818 3 5

) (

334338 2 818 3

)

]

58

1

+ +

+

+ − n + n − n

n x x x

x

[

(

220 3 4 818 3 3

) (

3220 2 818 1

)

]

609

1

+ +

+

+ − n + n − n

n y x y

On the Negative Pellian Equation y =110 x-29

67

[

(

8580 3 4 818 3 4

) (

38580 2 818 2

)

]

29

1

+ +

+

+ − n + n − n

n y x y

x

[

(

360140 3 4 818 3 5

) (

3360140 2 818 3

)

]

609

1

+ +

+

+ − n + n − n

n y x y

x

[

(

220 3 5 34338 3 3

) (

3220 3 34338 1

)

]

25549

1

+ +

+

+ − n + n − n

n y x y

x

[

(

8580 3 5 34338 3 4

) (

38580 3 34338 2

)

]

609

1

+ +

+

+ − n + n − n

n y x y

x

[

(

360140 3 5 34338 3 5

) (

3360140 3 34338 3

)

]

29

1

+ +

+

+ − n + n − n

n y x y

x

[

(

220 3 4 8580 3 3

) (

3220 2 8580 1

)

]

6380

1

+ +

+

+ − n + n − n

n y y y

y

[

(

220 3 5 360140 3 3

) (

3220 3 360140 1

)

]

267960

1

+ +

+

+ − n + n − n

n y y y

y

[

(

8580

_{3 5}

360140

_{3 4}

) (

3

8580

₃

360140

₂

)

]

6380

1

+ +

+

+

−

n

+

n

−

n

n

y

y

y

y

2.4. Each of the following expressions represents a bi-quadratic integer

(

) (

)

[

47444 1044 4818 18 6728

]

,

58

1 ₂

2 1

5 4 4

4

2 x n+ − x n+ + xn+ − xn+ −

(

) (

)

[

83647368 438948 434338 18 11868192

]

,

2436

1 ₂

3 1

6 4 4

4

2 x n+ − x n+ + xn+ − xn+ −

(

) (

)

[

6380 522 4220 18 1682

]

,

29

1 ₂

1 1

4 4 4

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

5225220 10962 48580 18 741762

]

, 609

1 ₂

2 1

5 4 4

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

9201216860 459882 4360140 18 130529841

]

, 25549

1 ₂

3 1

6 4 4

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

1991604 47444 434338 818 6782

]

,

58

1 ₂

3 2

6 4 5

4

2 x n+ − x n+ + xn+ − xn+ −

(

) (

)

[

133980 498162 4220 818 741762

]

,

609

1 ₂

1 2

4 4 5

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

248820 23722 48580 818 1682

]

,

29

1 ₂

2 2

5 4 5

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

219325260 498162 4360140 818 741762

]

, 609

1 ₂

3 2

6 4 5

4

68

(

) (

)

2

4 6 4 4 3 1

2

5620780

877301562

4 220

34338

1

,

25549

_1305502802

n n n n

x

₊

y

₊

x

₊

y

₊



₋

₊

₋







−









(

) (

)

2

4 6 4 5 3 2

2

5225220

20911842

4 8580

34338

1

,

609

₇₄₁₇₆₂

n n n n

x

₊

y

₊

x

₊

y

₊



₋

₊

₋







−









(

) (

)

[

10444060 995802 4360140 34338 1682

]

, 29

1 ₂

3 3

6 4 6

4

2 x n+ − y n+ + xn+ − yn+ −

(

) (

)

[

1403600 54740400 4220 8580 81408800

]

, 6380

1 ₂

1 2

4 4 5

4

2 y n+ − y n+ + yn+ − yn+ −

(

) (

)

2

4 6 4 4 3 1

2

58951200

9650311440

4 220

360140

1

,

267960

_14360512320

n n n n

y

₊

y

₊

y

₊

y

₊



₋

₊

₋







−









(

) (

)

2

4 6 4 5 3 2

2

54740400

2297693200

4 8580

360140

1

.

6380

_81408800

n n n n

y

₊

y

₊

y

₊

y

₊



₋

₊

₋







−









3. Remarkable observations

3.1. Employing the linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbola which are presented in the table2 below:

Table 2: Hyperbola S.

N o

Hyperbola

(

X ,

n

Y

n

)

1

110 1480160

2 2_{− =}

n n Y X

1 2

2 1

8580 220

18 818

+ +

+ +

− =

− =

n n

n

n n

n

x x

Y

x x

X

2 ₁₁₀ 2 ₋ 2 _=2611002240

n n Y X

1 3

3 1

360140 220

18 34338

+ +

+ +

− =

− =

n n

n

n n

n

x x

Y

x x

X

3 ₁₁₀ 2₋ 2 _=370040

n n Y X

1 1

1 1

1980 220

18 220

+ +

+ +

− =

− =

n n

n

n n

n

x y

Y

y x

X

4 ₁₁₀ 2₋ 2 _=163187640

n n Y X

1 2

2 1

89980 220

18 8580

+ +

+ +

− =

− =

n n

n

n n

n

x y

Y

y x

X

5 ₁₁₀ 2₋ 2_{=287210616440} n

n Y X

1 3

3 1

3777180 220

18 360140

+ +

+ +

− =

− =

n n

n

n n

n

x y

Y

y x

X

6 ₁₁₀ 2₋ 2 _=1480160

n n Y X

2 3

3 2

360140 8580

818 34338

+ +

+ +

− =

− =

n n

n

n n

n

x x

Y

x x

On the Negative Pellian Equation y =110 x-29

69 7 ₁₁₀ 2₋ 2 _=163187640

n n Y X 2 1 1 2 1980 8580 818 220 + + + + − = − = n n n n n n x y Y y x X

8 ₁₁₀ 2₋ 2 _=370040

n n Y X 2 2 2 2 89980 8580 818 8580 + + + + − = − = n n n n n n x y Y y x X

9 ₁₁₀ 2₋ 2 _=163187640

n n Y X 2 3 3 2 3777180 8580 818 360140 + + + + − = − = n n n n n n x y Y y x X 1

0 110 287210616440

2

2_{− =}

n n Y X 3 1 1 3 1980 360140 34338 220 + + + + − = − = n n n n n n x y Y y x X 1

1 110 163187640

2 2_{− =}

n n Y X 3 2 2 3 89980 360140 34338 8580 + + + + − = − = n n n n n n x y Y y x X 1

2 110 370040

2 2_{− =}

n n Y X 3 3 3 3 3777180 360140 34338 360140 + + + + − = − = n n n n n n x y Y y x X 1

3 110 17909936000

2 2_{− =}

n n Y X 2 1 1 2 1980 89980 8580 220 + + + + − = − = n n n n n n y y Y y y X 1

4

110

3159312710

4000

2

2

₋

₌

n n

Y

X

3 1 1 3 1980 3777180 360140 220 + + + + − = − = n n n n n n y y Y y y X 1

5

110

1790993600

0

2

2

₋

₌

n n

Y

X

3 2 2 3 89980 3777180 360140 8580 + + + + − = − = n n n n n n y y Y y y X

3.2. Employing the linear combination 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: Parabola S.

No

Parabola

(

X ,n Yn

)

1

1480160 6380X_n−Y_n2 =

1 2 3 2 2 2

8580

220

116

18

818

+ + + +

−

=

+

−

=

n n n n n n

x

x

Y

x

x

X

2 2611002240 267960X_n−Y_n2 =

1 3 4 2 2 2 360140 220 4872 18 34338 + + + + − = + − = n n n n n n x x Y x x X 3 370040 3190X_n−Y_n2=

1 1 2 2 2 2 1980 220 58 18 220 + + + + − = + − = n n n n n n x y Y y x X 4 163187640 66990X_n −Y_n2 =

70 5 _{2810390 −} 2 _{=287210616440}

n n Y X 1 3 4 2 1 2 3777180 220 51098 18 360140 + + + + − = + − = n n n n n n x y Y y x X

6 _{6380 −} 2 _=1480160

n n Y X 2 3 4 2 3 2 360140 8580 116 818 34338 + + + + − = + − = n n n n n n x x Y x x X

7 _{66990 −} 2 _=163187640

n n Y X 2 1 2 2 3 2 1980 8580 1218 818 220 + + + + − = + − = n n n n n n x y Y y x X

8 _{3190 −} 2 _=370040

n n Y X 2 2 3 2 3 2 89980 8580 58 818 8580 + + + + − = + − = n n n n n n x y Y y x X

9 _{66990 −} 2 _=163187640 n n Y X 2 3 4 2 3 2 3777180 8580 1218 818 360140 + + + + − = + − = n n n n n n x y Y y x X

10 _{281039 −} 2 _{=287210616440} n n Y X 3 1 2 2 4 2 1980 360140 51098 34338 220 + + + + − = + − = n n n n n n x y Y y x X

11 _{66990 −} 2 _=163187640

n n Y X 3 2 3 2 4 2 89980 360140 1218 34338 8580 + + + + − = + − = n n n n n n x y Y y x X

12 _{3190 −} 2 _=370040

n n Y X 3 3 4 2 4 2 3777180 360140 58 34338 360140 + + + + − = + − = n n n n n n x y Y y x X 13 0 1790993600 2

701800X_n−Y_n =

2 1 2 2 3 2 1980 89980 12760 8580 220 + + + + − = + − = n n n n n n y y Y y y X 14 4000 3159312710 29475600X_n−Y_n2 =

3 1 2 2 4 2 1980 3777180 535920 360140 220 + + + + − = + − = n n n n n n y y Y y y X 15 0 1790993600 701800X_n −Y_n2=

3 2 3 2 4 2 89980 3777180 12760 360140 8580 + + + + − = + − = n n n n n n y y Y y y X 4. Conclusion

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

represented by the negative pell equationy2 =110x2−29. As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of negative pell equations and determine their integer solutions along with suitable properties.

On the Negative Pellian Equation y =110 x-29

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