Observation on the Binary Quadratic Equation

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Observation on the Binary Quadratic Equation

0

,

4

105

2

2







x

t

y

t

G. Sumathi1, A. Prathiba2

1

Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India.

2

PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India.

Abstract: The binary quadratic equation is considered and a few interesting properties among the solutions are presented. Keywords:Binary quadratic, integral solutions, Generalized Fibonacci sequences, Generalized Lucas Sequences.

I. INTRODUCTION

The Binary quadratic equation of the form

y

2



Dx

2



1

, where D is a non-square positive integer has been studied by various Mathematicians for its non-trivial integral solutions when D takes different integral values [1-5]. In this context one may also refer [4,5]. These results have motivated us to search for the integral solutions of yet another binary quadratic equation representing a hyperbola. A few interesting properties among the solutions are presented.

II. NOTATIONS

1)

t

m,n : Polygonal number of rank

n

with size

m

2)

P

_nm : Pyramidal number of rank

n

with size

m

3)

Pr

_n : Pronic number of rank

n

4)

S

_n : Star number of rank

n

5)

Ct

m,n : Centered Pyramidal number of rank

n

with size

m

6)

GF

_n



k

,

s



: Generalized Fibonacci sequence number of rank n

7)

GL

_n



k

,

s



: Generalized Lucas sequence number of rank n

III. METHODOFANALYSIS

The binary non-homogeneous quadratic Diophantine equation represents a hyperbola to be solved for its non-zero integral solutions is

2 105 2 4, 0

 

 x t

y t

(1)

The smallest positive integer solution



x0,y0



of (1) is

 

t

x0 42 ,

 

t

y0412 (2)

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

y2 105x21 (3)

Applying the Brahmagupta lemma between the solutions (x0,y0) and (x~n,~yn), the other integer solutions of (1) are given by

 

 

n t

n t

n

f

g

x

105

2

2

41

2

2

4

1







 

 

n

t n

t

n

f

g

y

105

2

2

420

2

2

41

1





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A. A Few Numerical Examples Are Given In The Following Table 1

Tabe1: Examples

B. A Few Interesting Properties Are Given Below

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

0 2 164

2xn3 xn2 xn1 , n1,0,1...

0 2 164

2yn3 yn2 yn1 , n1,0,1...

A few interesting relations among the solutions are given below:

a) x_n382x_n2x_n10

b) 4y_n1x_n241x_n10

c) 4y_n241x_n2x_n10

d) 4yn33361xn241xn10

e) 328y_n1x_n33361x_n10

f)

8

y

n2



x

n3



x

n1



0

g) 328y_n33361x_n3x_n10

h) y_n241y_n1420x_n10

i) y_n33361y_n134440x_n10

j) 41y_n33361y_n2420x_n10

k) 4y_n141x_n33361x_n20

l) 4yn2xn341xn20

m)

4

y

n3



41

x

n3



x

n2



0

n) 41y_n2y_n1420x_n20

o) y_n341y_n2420x_n20

p) 3361y_n241y_n1420x_n30

q) 3361y_n3y_n134440x_n30

r) y_n3y_n1840x_n20

s) 41y_n3y_n2420x_n30

t) yn1yn382yn20

n x_n1 yn1

-1

 

t

2

4 41

 

2t

0

 

t

2

328 3361

 

2t

1

 

t

2

26892 275561

 

2t

2

 

t

2

2204816

 

t

953

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2) Each Of The Following Is A Nasty Number

a)

 

t



123x n 10083xn 12

 

2t



2 1

2 2 3

2    

b)

 

t



3xn 20163xn 8

 

2t



2 2

1

2 2 4

2    

c)

 

t



492y n 5040x n 12

 

2t



2 1

2 2 2

2    

d)

 

t



12y n 10080xn 12

 

2t



2 1

2 2 3

2    

e)

 

t



492y n 33883920xn 40332

 

2t



2 3361

1

2 2 4

2    

f)

 

t



10083xn 826683xn 12

 

2t



2 1

3 2 4

2    

g)

 

t



40332y n 5040xn 492

 

2t



2 41

1

3 2 2

2    

h)

 

t



40332y n 68880xn 12

 

2t



2 1

3 2 3

2    

i)

 

t



40332yn 33883920x n 492

 

2t



2 41

1

3 2 4

2   

j)

 

t



3306732y n 5040xn 40332

 

2t



2 3361

1

4 2 2

2    

k)

 

t



3306732y n 413280xn 492

 

2t



2 41

1

4 2 3

2    

l)

 

t



3306732yn 33883920xn 12

 

2t



2 1

4 2 4

2    

m)

 

t



34440yn 420y n 420

 

2t



2 35

1

3 2 2

2    

n)

 

t



40338yn 6yn 492

 

2t



2 41

1

4 2 2

2    

o)

 

t



80676y n 164y n 12

 

2t



2 1

4 2 3

2    

3) Each Of The Following Is A Cubical Integer

a)

 



41 3 4 3361 3 3 123 2 10083 1



2 2

1

 



  n  n  n

n

t x x x x

b)

 



3 5 6721 3 3 3 3 20163 1



2 4

1

 



  n  n  n

n

t x x x x

c)

 



82 3 3 840 3 3 246 1 2520 1



2 1

 



  n  n  n

n

t y x y x

d)

 



2 3 4 1680 2 2 6 2 5040 1



2 1

 



  n  n  n

n

t y x y x

e)

 



82 3 5 5647320 3 3 246 3 16941960 1



2 3361

1

 



  n  n  n

n

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f)

 



3361 3 5 275561 3 4 10083 3 826683 2



2 2

1

 



  n  n  n

n

t x x x x

g)

 



6722 3 3 840 3 4 20166 1 2520 2



2 41

1

 



  n  n  n

n

t y x y x

h)

 



6722 3 4 68880 3 4 20166 2 206640 2



2 2

1

 



  n  n  n

n

t y x y x

i)

 



6722 3 5 647320 3 4 20166 3 16941960 2



2 41

1

 



  n  n  n

n

t y x y x

j)

 



551122 3 3 840 3 5 1653366 1 2520 3



2 3361

1

 



  n  n  n

n

t y x y x

k)

 



551122 3 4 68880 3 5 1653366 2 206640 3



2 41

1

 



  n  n  n

n

t y x y x

l)

 



551122 3 5 5647320 3 5 1653366 3 16941960 3



2 1

 



  n  n  n

n

t y x y x

m)

 



34440 3 3 420 3 4 103320 1 1260 2



2 210

1

 



  n  n  n

n

t y y y y

n)

 



6723 3 3 3 5 20169 1 3 3



2 41

1

  

  n  n  n n

t y y y y

o)

 



13446 3 4 164 3 5 40338 2 492 3



2 1

 



  n  n  n

n

t y x y y

4) Each of the following expression is a biquadratic integer

a)

 

t



41xn 3361xn 164xn 13444x n 12

 

2t



2 2

1

4 2 3

2 4 4 5

4        

b)

 

t



x n 6721xn 4xn 26884xn 24

 

2t



2 4

1

2 2 4

2 4 4 6

4        

c)

 

t



82y n 840xn 328yn 3360xn 6

 

2t



2 1

2 2 2

2 4

4 4

4        

d)

 

t



2y n 1680xn 8yn 6720xn 6

 

2t



2 1

2 2 3

2 4 4 5

4        

e)

 

t



82yn 5647320x n 328y n 22589280xn 20166

 

2t



2 3361

1

2 2 4

2 4

4 6

4        

f)

 

t



3361x n 275561xn 13444x n 1102244x n 12

 

2t



2 2

1

3 2 4

2 5

4 6

4        

g)

 

t



6722yn 840xn 26888y n 3360xn 246

 

2t



2 41

1

3 2 2

2 5

4 4

4        

h)

 

t



6722y n 68880xn 26888y n 275520xn 6

 

2t



2 1

3 2 3

2 5

4 5

4        

i)

 

t



6722yn 56473270xn 26888yn 22589280x n 246

 

2t



2 41

1

3 2 4

2 5

4 6

4        

j)

 

t



551122y n 840x n 2204488y n 33604x n 20166

 

2t



2 3361

1

4 2 2

2 6

4 4

4        

k)

 

t



551122y n 68880xn 2204488yn 275520xn 246

 

2t



2 41

1

4 2 3

2 6

4 5

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l)

 

t



551122yn 5647320xn 2204488y n 22589280xn 6

 

2t



2 1 4 2 4 2 6 4 6

4        

m)

 

t



34440y n 420y n 137760y n 1680yn 1260

 

2t



2 210 1 3 2 2 2 5 4 4

4        

n)

 

t



6723yn y n 26892y n 4yn 246

 

2t



2 41 1 4 2 2 2 6 4 4

4        

o)

 

t



13446y n 164yn 53784y n 656yn 6

 

2t



2 1 4 2 3 2 6 4 5

4        

5) Each of the following expression is a quintic integer

a)

 

                     1 2 1 2 3 3 4 3 5 5 6 5 16805 205 50415 615 16805 205 3361 41 2 2 1 n n n n n n n n t x x x x x x x x b)

 

                     1 3 1 3 3 3 5 3 5 5 7 5 33605 5 100815 15 33605 5 6721 2 4 1 n n n n n n n n t x x x x x x x x c)

 

                     1 1 1 1 3 3 3 3 5 5 5 5 4200 410 12600 1230 4200 410 840 82 2 1 n n n n n n n n t x y x y x y x y d)

 



























        1 2 1 2 2 2 4 3 5 5 6 5

8400

10

25200

30

8400

10

1680

2

2

1

n n n n n n n n t

x

y

x

y

x

y

x

y

e)

 

                     1 3 1 3 3 3 5 3 5 5 7 5 28236600 410 84709800 1230 28236600 410 5647320 82 2 3361 1 n n n n n n n n t x y x y x y x y f)

 

                     2 3 2 3 4 3 5 3 6 5 7 5 1377805 16805 4133415 50415 1377805 16805 275561 3361 2 2 1 n n n n n n n n t x x x x x x x x g)

 

                     2 1 2 1 4 3 3 3 6 5 5 5 3225 193555 9675 58065 3225 19355 6722 2 41 1 n n n n n n n n t x y x y x y x y h)

 

                     2 2 2 2 4 3 4 3 6 5 6 5 344400 33610 100830 100830 344400 33610 68880 6722 2 1 n n n n n n n n t x y y y x y x y i)

 

                     2 3 2 3 4 3 5 3 6 5 7 5 28236600 33610 84709800 100830 28236600 33610 5647320 6722 2 41 1 n n n n n n n n t x y x y x y x y j)

 

                     3 1 3 1 5 3 3 3 7 5 5 5 4200 2755610 12600 8266830 4200 2755610 840 551122 2 3361 1 n n n n n n n n t x y x y x y x y k)

 

                     3 2 3 2 5 3 4 3 7 5 6 5 34440 2755610 1033200 8266830 344400 2755610 68880 551122 2 41 1 n n n n n n n n t x y x y x y x y l)

 

                     3 3 3 3 5 3 5 3 7 5 7 5 2823660 2755610 84709800 8266830 28236600 27556100 5647320 551122 2 1 n n n n n n n n

t _{x y x}

y x y x y m)

 

                     2 1 2 1 4 3 3 3 6 5 5 5 2100 172200 6300 516600 2100 172200 420 34440 2 210 1 n n n n n n n n

t _{y y y}

y y y y y n)

 

                     3 1 3 1 5 3 3 3 7 5 5 5 5 33615 15 100845 5 33615 6723 2 41 1 n n n n n n n n t y y y y y y y y o)

 

                     3 2 3 2 5 3 4 3 7 5 6 5 820 67230 2460 201690 820 67230 164 13446 2 1 n n n n n n n n

t _{y y y}

y y

y y

956

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6) Remarkable Observations

a) Employing linear combinations among the solutions of (1.1), one may generate integer solutions for other choices of hyperbola which are presented in table below.

S.No Hyperbola



X_n,Y_n



1

 

t

n

n Y

X2420 2 164

    

  

 

 

 

2 1

1 2

2 164

3361 41

n n

n n

x x

x x

2

 

t

n

n Y

X 1680 1075844

1681 2 2

 

   

 

 

 

 

3 1

1 3

6723 6721

n n

n n

x x

x x

3

 

t

n

n Y

X2105 2 44

   

 

 

 

 

1 1

1 1

8 82

840 82

n n

n n

y x

x y

4

 

t

n

n Y

X 105 67244

1681 2 2

   

 

 

 

 

2 1

1 2

8 6722

1680 2

n n

n n

y x

x y

5

 

t

n

n Y

X2105 2 451852844

   

 

 

 

 

3 1

1 3

8 551122

5647320 82

n n

n n

y x

x y

6

_{ }

t

n

n Y

X2420 2 164

   

 

 

 

 

3 2

2 3

164 13446

275561 3361

n n

n n

x x

x x

7

 

t

n n

Y

X

2



105

2



6724

4

   

 

 

 

 

1 2

2 1

656 82

840 6722

n n

n n

y x

x y

8

 

t

n

n Y

X2 105 2 44

 

   

 

 

 

 

2 2

2 2

656 6722

68880 6722

n n

n n

y x

x y

9

 

t

n

n Y

X2105 267244

   

 

 

 

 

3 2

2 3

656 551122

5647320 6722

n n

n n

y x

x y

10

 

t

n

n Y

X2105 2451852844

   

 

 

 

 

1 3

3 1

53784 82

840 551122

n n

n n

y x

x y

11

 

t

n

n Y

X2105 2 67244

   

 

 

 

 

2 3

3 2

53784 6722

68880 551122

n n

n n

y x

x y

12

 

t

n

n Y

X2105 244

   

 

 

 

 

3 3

3 3

53784 551122

5647320 551122

n n

n n

y x

x y

13

 

t

n

n Y

X2105 21764004

   

 

 

 

 

1 2

2 1

3361 41

420 34440

n n

n n

y y

y y

14

 

t

n

n Y

X 35301 2372227204

35280 2 2

 

   

 

 

 

 

1 3

3 1

6721 6723

n n

n n

y y

y y

15

 

t

n

n Y

X 21 352804

8820 2 2 

   

 

 

 

 

2 3

3 2

275561 3361

164 13446

n n

n n

y y

957

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b) Employing linear combinations among the solutions of (1.1), one may generate integer solutions for other choices of Parabola which are presented in table below.

S.No Parabola



Xn,Yn



1

 

 

t

n n

t

Y

X 210 44

2  2

            3 2 2 2 2 2 3 2 2 164 3361 41 n n n n x x x x

2

 

 

t

n n

t

Y

X 420 134484

2

1681  2 

            4 2 2 2 2 2 4 2 6723 6721 n n n n x x x x

3

 

 

t

n n t

Y

X 105 44

2  2 

            2 2 2 2 2 2 2 2 68 82 840 82 n n n n y x x y

4

_{ }

_{ }

t

n n t

Y

X 105 33624

2

1681  2 

            3 2 2 2 2 2 3 2 8 6722 1680 2 n n n n y x x y

5

 

 

t

n n t

Y

X 105 225926424

2 3361 2               4 2 2 2 2 2 4 2 8 551122 5647320 82 n n n n y x x y

6

 

 

t

n n

t

Y

X 210 44

2 2               2 2 3 2 3 2 2 2 656 82 840 6722 n n n n x x x y

7

 

 

t

n n t

Y

X 105 33624

2 41 2               2 2 3 2 3 2 2 2 656 82 840 6722 n n n n y x x y

8

 

 

t

n n t

Y

X 105 24

2  2

            3 2 3 2 3 2 3 2 656 6722 68880 6722 n n n n y x x y

9

_{ }

_{ }

t

n n t

Y

X 105 33624

2

41  2 

            4 2 3 2 3 2 4 2 656 551122 5647320 6722 n n n n y x x y

10

 

 

t

n n t

Y

X 105 225926424

2

3361  2 

            2 2 4 2 4 2 2 2 53784 82 840 551122 n n n n y x x y

11

_{ }

_{ }

t

n n t

Y

X 105 33624

2

41  2 

            3 2 4 2 4 2 3 2 53784 6722 68880 551122 n n n n y x x y

12

 

 

t

n n t

Y

X 105 24

2  2

            4 2 4 2 4 2 4 2 53784 551122 5647320 551122 n n n n y x x y

13

 

 

t

n n t

Y

X 105 1764004

2

210  2 

            2 2 3 2 3 2 2 2 3361 41 420 34440 n n n n y y y y

14

 

 

t

n n t

Y

X 861 28929604

2

35280  2

            2 2 4 2 4 2 2 2 6721 6723 n n n n y y y y

15

 

 

t

n n t

Y

X 21 176404

2

8820  2

958

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c) Employing the linear combination among the solutions of (1), one may generate integer solutions for other choices of straight line which are presented in the table below:

S.No. Straight line



Y,X



1 Y 2X Y xn36721xn1

1

2 3361

41   

 x_n x_n

X

2 Y  X Y 82yn1840xn1

1

2 1680

2   

 y_n x_n X

3 Y 41X Y 6722yn1840xn2

2

2 68880

6722   

 yn xn

X

4 Y 3361X Y 82yn₃1680xn₁

3

3 5647320

551122   

 yn xn

X

5 Y  X Y 41xn23361xn1

2

3 275561

3361   

 xn xn

X

6 Y 210X Y 34440yn₁420yn₂

1

1 840

82   

 yn xn

X

7 Y 210X Y 34440yn₁420yn₂

3

2 164

13446   

 yn yn

X

8 Y  X Y 6722yn₃5647320xn₂

3 1

6723   

 yn yn

X

9 Y 2X Y3361xn3275561xn2

1

2 1680

2   

 y_n x_n X

10 Y 2X Y 176yn₁2yn₂

2

3 275561

3361   

 xn xn

X

11 Y 2X Y xn₃6721xn₁

3

3 5647320

551122   

 y_n x_n

X

12 Y  X Y 82yn1840xn1

3

2 164

13446   

 y_n y_n

X

13 Y41X Y 6723yn₁yn₃

1

2 1680

2   

 yn xn

X

14 Y 3361X Y 82yn₃5647320xn₁

3

2 164

13446   

 y_n y_n

X

15 YX Y 551122yn₂68880xn₃

3 1

6723   

 yn yn

X

d) The solutions in terms of special integer sequence namely, generalized Fibonacci sequence GFn



k,s



and Lucas sequence



k s



GLn , are exhibited below.

12

 

2 1



82,1



164

 

2 n1



82,1



t n

t

n GL GF

x

 

2



82, 1



1680

 

2



82, 1



2 41

1 1

1      

 n

t n

t

n GL GF

959

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e) Employing the solutions (1),each of the following among the special polygonal, pyramidal, star numbers, pronic numbers is a congruent to under modulo 4

i)

2

1 , 3

3 2

1 5

3 105 1 12

             

  

 

 x

x

y y

t p s

p

ii)

2 1

5 2

, 4

5

1 12 105 1

4

    

  

 

    

  

 x

x y

y

s p ct

p

iii)

2

, 4

5 2

2 , 3 3

2

1 4 105 3

    

  

 

       

 

x x

y y

ct p t

p

iv)

2

, 3 5 2

1 3

105 6

                

 x

x

y y

t p pr

p

v)

2

1 2 , 3 4

1 2

, 4

5

6 105 1 4

             

  

 



x x

x y

t p ct

p

vi)

2

2 , 3 3

2 2

1 3

2 3

105 1

36

             

  

 

 



y x

y y

t p s

p

vii)

2

2 , 3 3

2 2

1 5

3 105 1 12

             

  

 



 x

x

y y

t p s

p

viii)

2

1 , 4 5

1 2

3 3

3 2

105 4

             

  

  



x x

y y

t p p

pt

ix)

2 1 , 4

3 1 4

1 2

, 6

5

3 105 1

6

    

 

 _

     

  

 

 

x x x y

y

t p p ct

p

x)

2

, 3 5 2

1 2 , 3 4

1

105 6

                

 

x x

y y

t p t

p

REFERENCES

[1] Dickson.L.E, History of Theory of Numbers and Diophantine Analysis, Dover Publications, New York 2005.

[2] Mordell.L.J, Diophantine Equations, Academic Press, New York 1970.

[3] Carmicheal.R.D, The Theory of Numbers and Diophantine Analysis, Dover Publication, New York 1959.

[4] Gopalan .M.A, Sumathi.G, Vidhyalakshmi.S. “Observation on the hyperbola t

y x2 19 2 3



 ” , Scholars Journal of the Engineering and

Technology2014;2(2A):152-155.

[5] Gopalan .M.A, Sumathi.G, Vidhyalakshmi.S. “Observation on the hyperbola 2 26 2 1

  x

y , Bessel J.Math., Vol 4, Issue(1), 2014;21-25.

[6] On Special familes of Hyperbola, 2



4 2



2 2, 1

  

 t 

y k k

x The International Journal of Science and Technology, Vol 2, Issue(3), Pg.No.94-97, March

2014.

[7] Dr.G.Sumathi “Observation on the hyperbola 2 182 2 14



 x

y ”, Journal of Mathematics and Informatics, Vol 11, 73-81, 2017

[8] Dr.G.Sumathi “Observation on the Pell Equation 2 14 2 4

  x

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

2018

[9] Dr.G.Sumathi “Observation on the Equation 2 312 2 1



 x

y ” International Journal of Mathematics Trends and Technology, Vol 50, Issue 4, 31-34, Oct

2017

[10] Dr.G.Sumathi “Observation on the Hyperbola 2 150 2 16



 x

y ” International Journal of recent Trends in Engineering and Research, Vol 3, Issue 9,