Integral Points on the Ternary Quadratic Diophantine Equation ,

Integral Points on the Ternary Quadratic

Diophantine Equation

_y

2

_

₃₃

_x

2

_

₄

t

_,

_t

_

₀

Dr. G. Sumathi1, W. Princy Dona 2

1

Assistant Professor,PG and Research Department of Mathematics, Shrimati Indira Gandhi College Trichy-2, Tamil Nadu, India.

2

M.Phil Scholar,PG and Research Department of Mathematics, Shrimati Indira Gandhi College Trichy-2, Tamil Nadu, India.

Abstract: The binary quadratic equation = + representing hyperbola is considered for finding its integer solutions. A few interesting properties among the solutions are presented. Also, we present infinitely many positive integer solutions in terms of Generalized Fibonacci sequences of numbers, Generalized Lucas sequences of numbers.

Keywords: Binary quadratic integral solutions, generalized Fibonacci Sequences of numbers, generalized Lucas Sequences of numbers.

AMS Mathematics Subject Classification: 11D09 Notations

GF

_n

(

k

,

s

)

:

Generalized Fibonacci Sequences of rank n.

GL

n

(

k

,

s

)

:

Generalized Lucas Sequences of rank n.





















2

)

2

)(

1

(

1

,

m

n

n

t

_mn





6

]

)

5

(

)(

2

(

)

1

(

[

n

n

m

n

m

P

_nm











Pr

_n



n

(

n



1

)

1

2

)

1

(

,







mn

n

Ct

_mn

S

_n



6

n

(

n



1

)



1

I. INTRODUCTION

The binary quadratic equation of the form = + 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 [1,2,4]. In [3] infinitely many Pythagorean triangles in each of which hypotenuse is four times the product of the generators added with unity are obtained by employing the non-integral solutions of binary quadratic equation = 3 + 1. In [5] a, special Pythagorean triangle is obtained by employing the integral solutions of = 182 + 14. In [6] different pattern of infinitely many Pythagorean triangles are obtained by employing the non-integral solutions of = 14 + 4. In this context one may also refer [7,8]. These results have motivated us to search for the integral solutions of yet another binary quadratic equation = 33 + 4 representing a hyperbola. A few interesting properties among the solutions are presented. Employing the integral solutions of the equation under consideration a few patterns of Pythagorean triangles are obtained.

II. METHODS OF ANALYSIS

Consider the binary quadratic equation t

x

y

2



33

2



4

,

t



0

(1)

with least positive integer solutions is

,

)

2

(

4

0 t

x



y

₀



23

(

2

)

t

consider the Pell equation

1

33

2

2

_

_x

_

y

₍₂₎

whose general solution

(

~

x

n

,

~

y

n

)

is represented by

n n

n

n

g

y

f

x

2

1

~

,

33

2

1

~

_

_

in which,





1





1

33

4

23

33

4

23













n n

n

f

g

_n





23



4

33



n1





23



4

33



n1

,

n





1

,

0

,

1

,

2

...

Applying Brahmagupta lemma between the solutions of



x

₀

,

y

₀



and



~

x

_n

,

~

y

_n



, the general solutions of equation (1) are found to be

n

t n

t

n

f

g

y

2

(

2

)

33

2

)

2

(

23

1







n t

n t

n

f

g

x

33

2

)

2

(

23

)

2

(

2

1







The recurrence relations satisfied by the values of

x

_n1 and

y

_n1 are respectively.

0

46

_{2 1}

3











 n n

n

x

x

x

0

46

_{2 1}

3











 n n

n

y

y

y

[image:2.612.197.415.436.544.2]

A few numerical examples are presented in the table 1 below:

Table 1: Numerical examples

n

1



n

x

y

_n1

-1 4(2)t 23(2)t

0 184(2)t 1057(2)t 1 8460(2)t 48599(2)t 2 388976(2)t 2234497(2)t 3 17884436(2)t 102738263(2)t 4 822295080(2)t 4723725601(2)t

A. A Few Interesting Relations Between The Solutions Are Given Below 1)

x

_n3



1057

x

_n1



184

y

_n1



0

2)

23

x

_n2



x

_n1



4

y

_n2



0

3)

23

x

_n2



23

x

_n1



184

y

_n2



0

4)

1057

x

_n3



x

_n1



184

y

_n3



0

5)

1057

y

_n1



6072

x

_n1



y

_n3



0

6)

1057

y

_n2



132

x

_n1



23

y

_n3



0

7)

x

_n1



46

x

_n2



x

_n3



0

9)

4

y

_n3



x

_n2



23

x

_n3



0

10)

23

x

_n1



x

_n2



4

y

_n1



0

11)

23

y

_n2



132

x

_n2



y

_n1



0

12)

x

_n3



23

x

_n2



4

y

_n2



0

13)

y

_n3



132

x

_n2



23

y

_n2



0

14)

23

x

_n1



1057

x

_n2



4

y

_n3



0

15)

y

_n1



132

x

_n2



23

y

_n3



0

16)

23

y

_n1



132

x

_n3



1057

y

_n2



0

17)

23

y

_n3



132

x

_n3



y

_n2



0

18)

y

_n1



6072

x

_n3



1057

y

_n3



0

19)

132

x

_n1



23

y

_n1



y

_n2



0

20)

y

_n3



y

_n1



46

y

_n2



0

B. Each Of The Following Expression Represents A Nasty Number

1)



_{n n} t



t

276

x

12684

x

48

(

2

)

)

2

(

4

1

2 2 3

2 







2)



_{n n} t



t

276

x

583188

x

2208

(

2

)

)

2

(

184

1

2 2 4

2 







3)



_{n n} t



t

276

y

1584

x

12

(

2

)

2

1

2 2 2

2 







4)



_{n n} t



t

276

y

72864

x

276

(

2

)

)

2

(

23

1

2 2 3

2 







5)



_{n n} t



t

276

y

3350160

x

12684

(

2

)

)

2

(

1057

1

2 2 4

2 







6)



_{n n} t



t

12684

x

583188

x

48

(

2

)

)

2

(

4

1

3 2 4

2 







7)



_{n n} t



t

12684

y

1584

x

276

(

2

)

)

2

(

23

1

3 2 2

2 







8)



_{n n} t



t

12684

y

72864

x

12

(

2

)

2

1

3 2 3

2 







9)



_{n n} t



t

12684

y

3350160

x

276

(

2

)

)

2

(

23

1

3 2 4

2 







10)



_{n n} t



t

583188

y

1584

x

12684

(

2

)

)

2

(

1057

1

4 2 2

11)



_{n n} t



t

583188

y

72864

x

276

(

2

)

)

2

(

23

1

4 2 3

2 







12)



_{n n} t



t

583188

y

3350160

x

12

(

2

)

2

1

4 2 4

2 







13)



_{n n} t



t

2208

y

48

y

48

(

2

)

)

2

(

4

1

3 2 2

2 







14)



_{n n} t



t

101520

y

48

y

2208

(

2

)

)

2

(

184

1

4 2 2

2 







15)



_{n n} t



t

101520

y

2208

y

48

(

2

)

)

2

(

4

1

4 2 3

2 







C. Each Of The Following Expression Represents A Cubical Integer

1)



69

₂

3171

₁

23

_{3 4}

1057

_{3 3}



)

2

(

2

1

 







n



n



n

n

t

x

x

x

x

2)



3

₃

6339

_{1 3 5}

2113

_{3 3}



)

2

(

4

1

 







n



n



n

n

t

x

x

x

x

3)



138

₁

792

₁

46

_{3 3}

264

_{3 3}



2

1

 







n



n



n

n

t

y

x

y

x

4)



6

₂

1584

₁

2

_{3 4}

528

_{3 3}



2

1

 







n



n



n

n

t

y

x

y

x

5)



138

₃

1675080

₁

46

_{3 5}

558360

_{3 3}



)

2

(

1057

1

 







n



n



n

n

t

y

x

y

x

6)



3171

₃

145797

₂

1057

_{3 5}

48599

_{3 4}



)

2

(

2

1

 







n



n



n

n

t

x

x

x

x

7)



6342

₁

792

₂

2114

_{3 3}

264

_{3 4}



)

2

(

23

1

 







n



n



n

n

t

y

x

y

x

8)



6342

₂

36432

₂

2114

_{3 4}

12144

_{3 4}



2

1

 







n



n



n

n

t

y

x

y

x

9)



6342

₃

1675080

₂

2114

_{3 5}

558360

_{3 4}



)

2

(

23

1

 







n



n



n

n

t

y

x

y

x

10)



291594

₁

792

₃

97198

_{3 3}

264

_{3 5}



)

2

(

1057

1

 







n



n



n

n

t

y

x

y

x

11)



12678

₂

1584

₃

4226

_{3 4}

528

_{3 5}



2

1

 







n



n



n

n

t

y

x

y

x

12)



291594

₃

1675080

₃

97198

_{3 5}

1558360

_{3 5}



2

1

 







n



n



n

n

13)



276

₁

6

₂

92

_{3 3}

2

_{3 4}



2

1

  





n



n



n

n

t

y

y

y

y

14)



6345

₁

3

₃

2115

_{3 3 3 5}



)

2

(

23

1

  





n



n



n

n

t

y

y

y

y

15)



12690

₂

276

₃

4230

_{3 4}

92

_{3 5}



2

1

 







n



n



n

n

t

y

y

y

y

D. Each Of The Following Expression Represents A Bi-Quadratic Integer

1)



_{n n n n} t



t

92

x

4228

x

23

x

1057

x

12

(

2

)

)

2

(

2

1

4 4 5 4 2 2 3

2 















2)



_{n n n n} t



t

23

x

8452

x

x

2113

x

24

(

2

)

)

2

(

4

1

4 4 6 4 2 2 4

2 















3)



_{n n n n} t



t

184

y

1056

x

46

y

264

x

6

(

2

)

2

1

4 4 4 4 2 2 2

2 















4)



_{n n n n} t



t

8

y

2112

x

2

y

528

x

6

(

2

)

2

1

4 4 5 4 2 2 3

2 















5)



_{n n n n} t



t

1184

y

2233440

x

46

y

558360

x

6342

(

2

)

)

2

(

1057

1

4 4 6 4 2 2 4

2 















6)



_{n n n n} t



t

4228

x

194396

x

1057

x

48599

x

12

(

2

)

)

2

(

2

1

5 4 6 4 3 2 4

2 















7)



_{n n n n} t



t

8456

y

1056

x

2114

y

264

x

138

(

2

)

)

2

(

23

1

5 4 4 4 3 2 2

2 















8)



_{n n n n} t



t

8456

y

48576

x

2114

y

12144

x

6

(

2

)

2

1

5 4 5 4 3 2 3

2 















9)



_{n n n n} t



t

8456

y

2233440

x

2114

y

558360

x

138

(

2

)

)

2

(

23

1

5 4 6 4 3 2 4

2 















10)



_{n n n n} t



t

388792

y

1056

x

97198

y

264

x

6342

(

2

)

)

2

(

1057

1

6 4 4 4 4 2 2

2 















11)



_{n n n n} t



t

16904

y

2112

x

4226

y

528

x

6

(

2

)

2

1

6 4 5 4 4 2 3

2 















12)



_{n n n n} t



t

388792

y

2233440

x

97198

y

558360

x

6

(

2

)

2

1

6 4 6 4 4 2 4

2 















13)



t



n n

n n

t 368 y 8y 92 y 2y 6(2)

2 1 5 4 4 4 3 2 2

2        

14)



t



n n

n n

t 8460 y 4y 2115 y y 138(2)

) 2 ( 23 1 6 4 4 4 4 2 2

2        

15)



t



n n

n n

t

16920

y

368

y

4230

y

92

y

6

(

2

)

2

1

6 4 5 4 4 2 3

E. Each Of The Following Expression Represents A Quintic Integer

1)



10

₃

21130

₁

5

_{3 5}

10565

_{3 3 5 7}

2113

_{5 5}



)

2

(

4

1

    





n



n



n



n



n

n

t

x

x

x

x

x

x

2)



20

₃

42260

₁

10

_{3 5}

21130

_{3 3}

2

_{5 7}

4226

_{5 5}



)

2

(

8

1

    





n



n



n



n



n

n

t

x

x

x

x

x

x

3)



460

₁

2640

₁

230

_{3 3}

1320

_{3 3}

46

_{5 5}

264

_{5 5}



2

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

4)



20

₂

5280

₁

10

_{3 4}

2640

_{3 3}

2

_{5 6}

528

_{5 5}



2

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

5)



460

₃

5583600

₁

230

_{3 5}

2791800

_{3 3}

46

_{5 7}

558360

_{5 5}



)

2

(

1057

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

6)



10570

₃

485990

₂

5285

_{3 5}

242995

_{3 4}

1057

_{5 7}

48599

_{5 6}



)

2

(

2

1

    





n



n



n



n



n

n

t

x

x

x

x

x

x

7)



21140

₁

2640

₂

10570

_{3 3}

1320

_{3 4}

2114

_{5 5}

264

_{5 6}



)

2

(

23

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

8)



21140

₂

121440

₂

10570

_{3 4}

60720

_{3 4}

2114

_{5 6}

12144

_{5 6}



2

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

9)



21140

₃

5583600

₂

10570

_{3 5}

2791800

_{3 4}

2114

_{5 7}

558360

_{5 6}



)

2

(

23

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

10)



971980

₁

2640

₃

485990

_{3 3}

1320

_{3 5}

97198

_{5 5}

264

_{5 7}



)

2

(

1057

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

11)



84520

₂

5280

₃

21130

_{3 4}

2640

_{3 5}

4226

_{5 6}

528

_{5 7}



2

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

12)



971980

₃

5583600

₃

485990

_{3 5}

2791800

_{3 5}

97198

_{5 7}

558360

_{5 7}



2

1

    





n



n



n



n



n

n

t

y

x

y

x

y

x

13)



920

₁

20

₂

460

_{3 3}

10

_{3 4}

92

_{5 5}

2

_{5 6}



2

1

    





n



n



n



n



n

n

t

y

y

y

y

y

y

14)

_

_

7 5 5 5 5 3 3 3 3

1 10 10575 5 2115

21150 ) 2 ( 8 1     

  n  n  n  n  n

n

t y y y y y y

15)

_

_

7 5 6 5 5 3 4 3 3

2 920 21150 460 4230 92

42300 2 1     

  n  n  n  n  n

n

t y y y y y y

III. REMARKABLE OBSERVATIONS

1) The solutions of (1) in terms of special integers namely, generalized Fibonacci

GF

n and

Lucas

GL

_n are exhibited below.

)

1

,

46

(

)

2

(

92

)

1

,

46

(

)

2

(

2

_{1 1}

1













 t n t n

n

GL

GF

x

)

1

,

46

(

)

2

(

528

)

1

,

46

(

2

)

2

(

23

1 1

1













 n t n

t

n

GL

GF

2) Employing The 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 2 2

)

2

(

64

33

Y

t

X







46

x

n2



2114

x

n1

,

368

x

n1



8

x

n2



2 2 2 2

)

2

(

135424

33

Y

t

X







46

x

n3



97198

x

n1

,

16920

x

n1



8

x

n3



3 2 2 2

)

2

(

4

33

Y

t

X







46

y

n1



264

x

n1

,

46

x

n1



8

y

n1



4 2 2 2

)

2

(

2116

33

Y

t

X







46

y

n2



12144

x

n1

,

2114

x

n1



8

y

n2



5 2 2 2

)

2

(

4468996

33

Y

t

X





(

46

y

_n3



558360

x

_n1

,

97198

x

_n1



8

y

_n3)

6 2 2 2

)

2

(

64

33

t

Y

X







2114

x

n3



97198

x

n2

,

16920

x

n2



368

x

n3



7 2 2 2

)

2

(

2116

33

Y

t

X







2114

y

n1



264

x

n2

,

46

x

n2



368

y

n1



8 2 2 2

)

2

(

4

33

Y

t

X







2114

y

n2



12144

x

n2

,

2114

x

n2



368

y

n2



9 2 2 2

)

2

(

2116

33

t

Y

X







2114

y

n3



558360

x

n2

,

97198

x

n2



368

y

n3



10 2 2 2

)

2

(

4468996

33

Y

t

X







97198

y

n1



264

x

n3

,

46

x

n3



16920

y

n1



11 2 2 2

)

2

(

2116

33

Y

t

X







97198

y

n2



12144

x

n3

,

2114

x

n3



16920

y

n2



12 2 2 2

)

2

(

4

33

Y

t

X







97198

y

n3



558360

x

n3

,

97198

x

n3



16920

y

n3



13 2 2 2

)

2

(

2112

33

X



Y



t



368

y

n1



8

y

n2

,

46

y

n2



2114

y

n1



14 2 2 2

)

2

(

4468992

33

X



Y



t



16920

y

n1



8

y

n3

,

46

y

n3



97198

y

n1



15 2 2 2

)

2

(

2112

3) 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

₂

t

_X

_

₁₃₂

_Y

2

_

₈

₍

₄

₎

t





2 1

2 2 3

2

2114

,

368

8

46

x

_n



x

_n

x

_n



x

_n

2

₁₈₄

₍

₂

₎

t

_X

_

₃₃

_Y

2

_

₆₇₇₁₂

₍

₄

₎

t





3 1

2 2 4

2

97198

,

16920

8

46

x

_n



x

_n

x

_n



x

_n

3 t t

Y

X

33

2

(

4

)

2



2





46

y

2n2



264

x

2n2

,

46

x

n1



8

y

n1



4 t t

Y

X

33

1058

(

4

)

)

2

(

23



2





46

y

2n3



12144

x

2n2

,

2114

x

n1



8

y

n2



5 t t

Y

X

33

2234498

(

4

)

)

2

(

1057



2





46

y

2n4



558360

x

2n2

,

97198

x

n1



8

y

n3



6

₄

₍

₂

₎

t

_X

_

₃₃

_Y

2

_

₃₂

₍

₄

₎

t



₂₁₁₄

_{2 4}

₉₇₁₉₈

_{2 3}

_,

₁₆₉₂₀

₂

₃₆₈

₃



 







n n



n

n

x

x

x

x

7 t t

Y

X

33

1058

(

2

)

)

2

(

23



2





2114

y

2n2



264

x

2n3

,

46

x

n2



368

y

n1



8 t t

Y

X

33

2

(

4

)

2



2





2114

y

2n3



12144

x

2n3

,

2114

x

n2



368

y

n2



9

₂₃

₍

₂

₎

t

_X

_

₃₃

_Y

2

_

₁₀₅₈

₍

₄

₎

t





3 2

3 2 4

2

558360

,

97198

368

2114

y

_n



x

_n

x

_n



y

_n

10 t t

Y

X

33

2234498

(

4

)

)

2

(

1057



2





97198

y

2n2



264

x

2n4

,

46

x

n3



16920

y

n1



11 t t

Y

X

33

1058

(

4

)

)

2

(

23



2





97198

y

2n3



12144

x

2n4

,

2114

x

n3



16920

y

n2



12 t t

Y

X

33

2

(

4

)

2



2





97198

y

2n4



558360

x

2n4

,

97198

x

n3



16920

y

n3

13

₁₃₂

₍

₂

₎

t

_X

_

_Y

2

_

₁₀₅₆

₍

₄

₎

t





1 2

3 2 2

2

8

,

46

2114

368

y

_n



y

_n

y

_n



y

_n

14 t t

Y

X

2234496

(

4

)

)

2

(

6072



2





16920

y

2n2



8

y

2n4

,

46

y

n3



97198

y

n1



15 t t

Y

X

1056

(

4

)

)

2

(

4) Employing the following solution

(

x

,

y

)

, each of the following expressions among the special polygonal, pyramidal, star, pronic and centered polygonal number is congruent to under modulo 4.

2

1 3 2

2 , 3

3 2

Pr

198

3



































 



x x y

y

P

t

P

2

2 3

2 2

1 5

1

1188

1

12







































_



 x

x y

y

S

P

S

P

2

1 5 2

, 3

5

1

396





































 x

x y

y

S

P

t

P

2

, 6

5 2

, 4

5

198

1

2





































_x

x y

y

Ct

P

Ct

P

2

, 4

5 2

3 3

3

1

132

4





































 

x x

n n

Ct

P

P

Pt

IV. CONCLUSION

To conclude, one may search for other choices of positive Pell equations for finding their integer solutions with suitable properties.

REFERENCES

[1] L.E.Dickson, History of Theory of numbers, Vol.2, Chelsea Publishing Company, New York, (1952). [2] L.J.Mordell, Diophantine Equations, Academic Press, New York, (1970).

[3] M.A.Gopalan and G.Janaki, Observation On

y

2



3

x

2



1

, Acta Ciancia Indica, XXXIVM(2) (2008) 639-696. [4] L.E.Dickson, History of theory of numbers and Diophantine analysis, Vol 2, Dover publications, New York,(2005).

[5] Dr.G.Sumathi “Observations on the hyperbola”

y

2



182

x

2



14

,

Journal of mathematics and Informatics,Vol 11, 73-81, 2017.

[6] Dr.G.Sumathi “Observations on the Pell Equation

y

2



14

x

2



4

” International Journal of Creative Research Thoughts, Vol 6, Issue 1, Pp1074-1084,March 2018.

[7] Dr.G.Sumathi “Observations on the Equation

y

2



312

x

2



1

” International Journal of Mathematics Trends and Technology, Vol 50, Issue4, 31-34, Oct 2017.