On the Hyperbola $2x^2-3y^2=23$

Vol. 10, 2017, 1-9

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

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/jmi.v10a1

1

Journal of

On the Hyperbola 2x

2

-3y

2

=23

Sharadha Kumar1 and M.A.Gopalan2

1

Deptarment of Mathematics, SIGC, Trichy-620002, Tamilnadu, India. e-mail: [email protected]

2

Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India. e-mail: [email protected]

Received 15 November 2017; accepted 4 December 2017

Abstract. The hyperbola represented by the binary quadratic equation 2x2−3y2 =23 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, Parabola, Integral solutions, Pell equation. AMS Mathematics Subject Classification (2010): 11D09

1. Introduction

The binary quadratic Diophantine equations of the form ax2−by2 =N,

(

a,b,N≠0

)

are rich in variety and have been analyzed by many mathematicians for their respective integer solutions for particular values of a,bandN . In this context, one may refer [1-13].

This communication concerns with the problem of obtaining non-zero distinct integer solutions to the binary quadratic equation given by 2x2 −3y2 =23 representing hyperbola. A few interesting relations among its solutions are presented. 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.

2. Method of analysis

The binary quadratic equation representing hyperbola is given by 23

3

2x2 − y2 = (1)

Takingx= X +3T,y= X +2T (2) in (1), it simplifies to the equation

23 6 2

2 _{= −}

T

X (3)

2

1

,

2

₀

0

=

X

=

T

To obtain, the other solutions of (3), consider the pellian equation 1

6 2

2 _{= +}

T

X (4)

whose smallest positive integer solution is 5 ~ , 2 ~ 0

0 = X =

T

The general solution (T~_n,X~_n)of (4) is given by

(

5 2 6

)

, 0,1,2.... ~ 6 ~ 1 = + =

+ T + n

X_{n n} n (5)

Since, irrational roots occur in pairs, we have

(

5 2 6

)

, 0,1,2.... ~ 6 ~ 1 = − =

− T + n

X_{n n} n (6)

From (5) and (6), solving for X~_n,T~_n ,we have

(

) (

)

n

n n n f X 2 1 6 2 5 6 2 5 2 1

~ 1 1

=     _{+ + −} = + +

(

) (

)

n

n n n g T 6 2 1 6 2 5 6 2 5 6 2 1

~ 1 1

=     _{+ − −} = + +

Applying Brahmagupta lemma between the solutions

(

T

₀

,

X

₀

)

and (T~_n,X~_n), the general solution

(

T

_n+1

,

X

_n+1

)

of (3) is found to be

n n

n X T T X

T ₊₁ = ₀~ + ₀~

n n

n X X T T

X ₊₁ = ₀~ +6 ₀~

n n

n g f

T = +

⇒ ₊

6 2

1

1 (7)

n n

n f g

X 6

2 1

1 = +

+ (8)

Using (7) and (8) in (2) we have

n n

n n

n X T f g

x 6 2 15 2 7 3 ₁ 1

1 = + + + = +

+ (9)

n n

n n

n X T f g

y 6 7 2 5 2 ₁ 1

1 = + + + = +

+ (10)

Thus, (9) and (10) represent the integer solutions of the hyperbola (1). A few numerical examples are given in the following table 1.

Table 1: Examples

n

1

+

n

x

y

_n+1

On the Hyperbola 2x-3y =23

3 Recurrence relations for xand y are:

...

1

,

0

,

1

,

0

10

_{2 1}

3

−

+

+

+

=

=

−

+

x

x

n

x

_{n n n}

...

1

,

0

,

1

,

0

10

_{2 1}

3

−

+

+

+

=

=

−

+

y

y

n

y

_{n n n}

A few interesting relations among the solutions are given below:

•

x

_n+2

−

5

x

_n+1

−

6

y

_n+1

=

0

•

5

x

_n+2

−

x

_n+1

−

6

y

_n+2

=

0

•

49

x

_n+2

−

5

x

_n+1

−

6

y

_n+3

=

0

•

x

_n+3

−

49

x

_n+1

−

60

y

_n+1

=

0

•

x

_n+3

−

x

_n+1

−

12

y

_n+2

=

0

•

49

x

_n+3

−

x

_n+1

−

60

y

_n+3

=

0

•

49

x

_n+1

+

60

y

_n+1

−

x

_n+3

=

0

•

4

x

_n+1

+

5

y

_n+1

−

y

_n+2

=

0

•

40

x

_n+1

+

49

y

_n+1

−

y

_n+3

=

0

•

x

_n+1

+

6

y

_n+2

−

5

x

_n+2

=

0

•

4

x

_n+1

+

49

y

_n+2

−

5

y

_n+3

=

0

•

6

y

_n+3

+

5

x

_n+1

−

49

x

_n+2

=

0

•

y

_n+3

−

40

x

_n+1

−

49

y

_n+1

=

0

•

5

x

_n+3

−

49

x

_n+2

−

6

y

_n+1

=

0

•

x

_n+3

−

5

x

_n+2

−

6

y

_n+2

=

0

•

5

x

_n+3

−

x

_n+2

−

6

y

_n+3

=

0

•

49

x

_n+2

+

6

y

_n+1

−

5

x

_n+3

=

0

•

4

x

_n+2

+

y

_n+1

−

5

y

_n+2

=

0

•

y

_n+3

−

8

x

_n+2

−

y

_n+1

=

0

•

y

_n+1

+

8

x

_n+2

−

y

_n+3

=

0

•

4

x

_n+2

+

5

y

_n+2

−

y

_n+3

=

0

•

4

x

_n+3

+

5

y

_n+1

−

49

y

_n+2

=

0

•

40

x

_n+3

+

y

_n+1

−

49

y

_n+3

=

0

•

4

x

_n+3

+

y

_n+2

−

5

y

_n+3

=

0

•

5

y

_n+3

−

y

_n+2

−

4

x

_n+3

=

0

4

[

(

3 3 3 4

) (

1 2

)

]

5 53 3 5 53 23 1 + + +

+ − n + n − n

n x x x

x

[

(

525 3 3 5 3 5

) (

3525 1 5 3

)

]

300

1

+ +

+

+ − n + n − n

n x x x

x

[

(

28 3 3 30 3 3

) (

328 1 30 1

)

]

23

1

+ +

+

+ − n + n − n

n y x y

x

[

(

260 3 3 30 3 4

) (

3260 1 30 2

)

]

115

1

+ +

+

+ − n + n − n

n y x y

x

[

(

2572 3 3 30 3 5

) (

32572 1 30 3

)

]

1127

1

+ +

+

+ − n + n − n

n y x y

x

[

(

3150 3 4 318 3 5

) (

33150 2 318 3

)

]

138

1

+ +

+

+ − n + n − n

n x x x

x

[

(

28 3 4 318 3 3

) (

328 2 318 1

)

]

115

1

+ +

+

+ − n + n − n

n y x y

x

[

(

260 3 4 318 3 4

) (

3260 2 318 2

)

]

23

1

+ +

+

+ − n + n − n

n y x y

x

[

(

2572 3 4 318 3 5

) (

32572 2 318 3

)

]

115

1

+ +

+

+ − n + n − n

n y x y

x

[

(

28 3 5 3150 3 3

) (

328 3 3150 1

)

]

1127

1

+ +

+

+ − n + n − n

n y x y

x

[

(

260 3 5 3150 3 4

) (

3260 3 3150 2

)

]

115

1

+ +

+

+ − n + n − n

n y x y

x

[

(

2572 3 5 3150 3 5

) (

32572 3 3150 3

)

]

23

1

+ +

+

+ − n + n − n

n y x y

x

[

(

28 3 4 260 3 3

) (

328 2 260 1

)

]

92

1

+ +

+

+ − n + n − n

n y y y

y

[

(

28 3 5 2572 3 3

) (

328 3 2572 1

)

]

920

1

+ +

+

+ − n + n − n

n y y y

y

[

(

260 3 5 2572 3 4

) (

3260 3 2572 2

)

]

92

1

+ +

+

+ − n + n − n

n y y y

y

Each of the following expressions represents bi-quadratic integer:

a)

[

(

1219

115

) (

4

53

5

)

1058

]

23

1

2 2 1 5 4 4 4

2

x

n+

−

x

n+

+

x

n+

−

x

n+

−

b)

[

(

157500

1500

) (

4

525

5

)

180000

]

300

1

2 3 1 6 4 4 4

On the Hyperbola 2x-3y =23

5

c)

[

(

694

690

) (

4

28

30

)

1058

]

23

1

2 1 1 4 4 4 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

d)

[

(

29900

3450

) (

4

260

30

)

26450

]

115

1

2 2 1 5 4 4 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

e)

[

(

2898644

33810

) (

4

2572

30

)

2540258

]

1127

1

2 3 1 6 4 4 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

f)

[

(

434700

43884

) (

4

3150

318

)

38088

]

138

1

2 3 2 6 4 5 4

2

x

n+

−

x

n+

+

x

n+

−

x

n+

−

g)

[

(

3220

36570

) (

4

28

318

)

26450

]

115

1

2 1 2 4 4 5 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

h)

[

(

5980

7314

) (

4

260

318

)

1058

]

23

1

2 2 2 5 4 5 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

i)

[

(

295780

36570

) (

4

2572

318

)

26450

]

115

1

2 3 2 6 4 5 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

j)

[

(

31556

3550050

) (

4

28

3150

)

2540258

]

1127

1

2 1 3 4 4 6 4

2

x

n+

−

y

n+

+

x

n+

−

y

n+

−

k)

[

(

29900 362250

) (

4260 3150

)

26450

]

115 1 2 2 3 5 4 6 4

2 x n+ − y n+ + xn+ − yn+ −

l)

[

(

59156 75450

) (

42572 3150

)

1058

]

23 1 2 3 3 6 4 6 4

2 x n+ − y n+ + xn+ − yn+ −

m)

[

(

2576 23920

) (

428 260

)

16928

]

92 1 2 1 2 4 4 5 4

2 y n+ − y n+ + yn+ − yn+ −

n)

[

(

25760

2366240

) (

4

28

2572

)

1692800

]

920

1

2 1 3 4 4 6 4

2

y

n+

−

y

n+

+

y

n+

−

y

n+

−

o)

[

(

23920

236624

) (

4

260

2572

)

16928

]

92

1

2 2 3 5 4 6 4

2

y

n+

−

y

n+

+

y

n+

−

y

n+

−

Each of the following expressions represents Nasty number:

a)

[

276 318 _{2 2} 30 _{2 3}

]

23

1

+ + −

+ x n x n

b)

[

2760 3150 _{2 2} 30 _{2 4}

]

230

1

+ + −

+ x n x n

c)

[

276 168 _{2 2} 180 _{2 2}

]

23

1

+ + −

+ x n y n

d)

[

1380 1560 2 2 180 2 3

]

115

1

+ + −

6

e)

[

13524 15432 2 2 180 2 4

]

1127

1

+ + −

+ x n y n

f)

[

1656 18900 _{2 3} 1908 _{2 4}

]

138

1

+ + −

+ x n x n

g)

[

1380 168 2 3 1908 2 2

]

115

1

+ + −

+ x n y n

h)

[

276 1560 _{2 3} 1908 _{2 3}

]

23

1

+ + −

+ x n y n

i)

[

1380 15432 _{2 3} 1908 _{2 4}

]

115

1

+ + −

+ x n y n

j)

[

13524 168 _{2 4} 18900 _{2 2}

]

1127

1

+ + −

+ x n y n

k)

[

1380 1560 _{2 4} 18900 _{2 3}

]

115

1

+ + −

+ x n y n

l)

[

276 15432 _{2 4} 18900 _{2 4}

]

23

1

+ + −

+ x n y n

m)

[

1104 168 _{2 3} 1560 _{2 2}

]

92

1

+ + −

+ y n y n

n)

[

11040 168 _{2 4} 15432 _{2 2}

]

920

1

+ + −

+ y n y n

o)

[

1104 1560 _{2 4} 15432 _{2 3}

]

92

1

+ + −

+ y n y n

3. Remarkable observations

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

Table 2: Hyperbola

Sl.no Hyperbola

(

)

n n

Y

X ,

1 ₆ 2₋ 2 ₌₁₂₆₉₆

n

n Y

X

[

(

53

x

n+1

−

5

x

n+2

) (

,

14

x

n+2

−

130

x

n+1

)

]

2 ₆ 2₋ 2 _=1269600

n

n Y

X

[

(

525

x

n+1

−

5

x

n+3

) (

,

14

x

n+3

−

1286

x

n+1

)

]

3 ₆ 2₋ 2 ₌₁₂₆₉₆

n

n Y

X

[

(

28

x

n+1

−

30

y

n+1

) (

,

84

y

n+1

−

60

x

n+1

)

]

4 ₆ 2 ₋ 2 _=3174400

n

n Y

X

[

(

260

x

n+1

−

30

y

n+2

) (

,

84

y

n+2

−

636

x

n+1

)

]

5 ₆ 2 ₋ 2 _=30483096

n

n Y

X

[

(

2572

x

n+1

−

30

y

n+3

) (

,

84

y

n+3

−

6300

x

n+1

)

]

6 ₆ 2 ₋ 2 _=457056

n

n Y

X

[

(

3150

x

n+2

−

318

x

n+3

) (

,

780

x

n+3

−

7716

x

n+2

)

]

7 ₆ 2 ₋ 2 _=317400

n

n Y

On the Hyperbola 2x-3y =23

7 8 ₆ 2₋ 2 ₌₁₂₆₉₆

n

n Y

X

[

(

260

x

n+2

−

318

y

n+2

) (

,

780

y

n+2

−

636

x

n+2

)

]

9 ₆ 2 ₋ 2 _=317400

n

n Y

X

[

(

2572

x

n+2

−

318

y

n+3

) (

,

780

y

n+3

−

6300

x

n+2

)

]

10 ₆ 2 ₋ 2 _=30483096

n

n Y

X

[

(

28

x

n+3

−

3150

y

n+1

) (

,

7716

y

n+1

−

60

x

n+3

)

]

11 ₆ 2 ₋ 2 _=317400

n

n Y

X

[

(

260

x

n+3

−

3150

y

n+2

) (

,

7716

y

n+2

−

636

x

n+3

)

]

12 ₆ 2₋ 2 ₌₁₂₆₉₆

n

n Y

X

[

(

2572

x

n+3

−

3150

y

n+3

) (

,

7716

y

n+3

−

6300

x

n+3

)

]

13 ₆ 2 ₋ 2 _=203136

n

n Y

X

[

(

28

y

n+2

−

260

y

n+1

) (

,

636

y

n+1

−

60

y

n+2

)

]

14 ₆ 2₋ 2 _=20313600

n

n Y

X

[

(

28

y

n+3

−

2572

y

n+1

) (

,

6300

y

n+1

−

60

y

n+3

)

]

15 ₆ 2 ₋ 2 _=203136

n

n Y

X

[

(

260

y

n+3

−

2572

y

n+2

) (

,

6300

y

n+2

−

636

y

n+3

)

]

Employing linear combination among the solutions for other choices of parabola which are presented in table 3 below

8

3.3. Properties

(i) Let

p,

q

be two non-zero distinct integers such thatp>q>0, treat

p,

q

as the generators of the Pythagorean triangle T

(

α,β,γ

)

where

α

=

2

pq

,

β

=

p

2

−

q

2

,

γ

=

p

2

+

q

2

,

p

>

q

>

0

Treating

p

=

x

_n+1

+

y

_n+1

,

q

=

x

_n+1, it is observed that T

(

α,β,γ

)

is satisfied by the following relations:

23 2 3

α

−

β

−

γ

=

γ

β

α

+ − =

P A

4

1 1

2 =xn₊ yn₊

P A

whereA,P represents the area and perimeter of T

(

α,β,γ

)

.

(ii) Let M, N be two non-zero distinct positive integers.

Consider

2 1 ,

2

1 1

1− ₌ −

= n+ yn+

M x

N

It is observed that 3 3

2t_3,N − t_3,M = , t_3,M = Triangular number of rank M.

4. Conclusion

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

equation, represented by hyperbola is given by 2x2 −3y2 =23. As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of equations and determine their integer solutions along with suitable properties.

REFERENCES

1. R.D.Carmichael, The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1950).

2. L.E.Disckson, History of theory of Numbers, Vol.II, Chelsea Publishing co., New York (1952).

3. L.J.Mordell, Diophantine Equations, Academic Press, London (1969).

4. M.A.Gopalan and R.Anbuselvi, Integral solutions of

4

ay

2

−

(

a

−

1

)

x

2

=

3

a

+

1

Acta Ciencia Indica, XXXIV (1) (2008) 291-295. 5. Gopalan et al., Integral points on the hyperbola

(

a

+

2

)

x

2

−

ay

2

=

4

a

(

k

−

1

)

+

2

k

2

,

a

,

k

>

0

, Indian Journal of Science, 1(2) (2012) 125-126.

6. M.A.Gopalan, S.Devibala and R.Vidhyalakshmi, Integral points on the hyperbola 5

3

On the Hyperbola 2x-3y =23

9

7. S.Vidhyalakshmi, et al., Observations on the hyperbola

ax

2

−

(

a

+

1

)

y

2

=

3

a

−

1

Discovery, 4(10) (2013) 22-24.

8. K.Meena, M.A.Gopalan and S.Nandhini, On the binary quadratic Diophantine equation

y

2

=

68

x

2

+

13

, International Journal of Advanced Education and Research, 2(1) (2017) 59-63.

9. K.Meena, S.Vidhyalakshmi and R.Sobana Devi, On the binary quadratic equation

32

7

2

2

=

+

x

y

, International Journal of Advanced Science and Research, 2(1) (2017) 18-22.

10. K.Meena, M.A.Gopalan, S.Hemalatha, On the hyperbola

y

2

=

8

x

2

+

16

, National Journal of Multidisciplinary Research and Developement, 2(1) (2017) 1-5.

11. M.A.Gopalan, K.K.Viswanathan, G.Ramya, On the Positibe Pell equation

13

12

2

2

=

+

x

y

, International Journal of Advanced Education and Research, 2(1) (2017) 4-8.

12. K.Meena, M.A.Gopalan and V.Sivaranjani, On the Positive Pell equation

33

102

2

2

=

+

x

y

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

13. K.Meena, S.Vidhyalakshmi and N.Bhuvaneswari, On the binary quadratic Diophantine equation

_y

2

=

10

_x

2

+

24

, International Journal of Multidisciplinary Education and Research, 2(1) (2017) 34-39.

14. Gopalan et al., Integral points on the hyperbola

(

a+2

)

x2 −ay2 =4a

(

k−1

)

+2k2,a,k >0, Indian Journal of Science, 1(2) (2012) 125-126.

15. M.A.Gopalan and V.Geetha, Observations on some special pellian equations, Cayley J. Math., 2(2) (2013) 109-118.

16. M.A.Gopalan,S.Vidhyalakshmi and A.Kavitha,On the integer solutions of binary