IRJET- On the Binary Quadratic Diophantine Equation Y2=272x2+16

International Research Journal of Engineering and Technology (IRJET)

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ON THE BINARY QUADRATIC DIOPHANTINE EQUATION y2 =272x2 + 16 Dr. G. Sumathi1, S. Gokila2 1Assistant

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

2PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

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Abstract – The

binary y  272 x  16 is considered properties among the solutions the integral solutions of considerations a few patterns of observed.

2

2

quadratic equation and a few interesting are presented .Employing the equation under Pythagorean triangle are

whose smallest positive integer solutions of

x0  8 , y0  132

~ x0  2 , ~ y0  33

1. INTRODUCTION

whose general solution ~ xn , ~ yn  of (3) is given by,

The 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 [1-5]. In this context one may also refer [4,10]. These results have motivated us to search for the integral solutions of yet another binary quadratic equation y 2  272 x 2  16 representing a hyperbola .A few interesting properties among the solution are presented.

Employing the integral solutions of the equation consideration a few patterns of Pythagorean triangles are obtained.

1 1 ~ xn  gn , ~ yn  f n 272 2 where,

f n  (33  2 272 ) n1  (33  272) n1 g n  (33  2 272 ) n1  (33  2 272 ) n1 Applying Brahmagupta lemma between x0 , y0  and ~ xn , ~ yn  the other integer solution of (1) are given by,

1.1 Notations

66

xn1  4 f n 

272

t m,n : Polygonal number of rank n with size m

Pn m : Pyramidal number of rank n with size m

yn1  9 f n 

Prn : Pronic number of rank n S n : Star number of rank n Ctm,n : Centered Pyramidal number of rank n with

20

(5)

gn

272 xn1  4 272 f n  66 g n

(6)

Replace n by n  1 in (6),we get

2. METHOD OF ANALYSIS

272 xn 2  4 272 f n1  66 g n1  4 272 (33 f n  2 272 g n )  66(33g n  2 272 f n )

Consider the binary quadratic Diophantine equation is

272 xn 2  264 272 f n  4354 g n

(1)

Replace

|

40

(4)

gn

Therefore (4) becomes

size m GFn (k , s) : Generalized Fibonacci sequence of rank n GLn (k , s) : Generalized Lucas sequence of rank n

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(2)

To obtain the other solutions of (1),Consider pellian equation is (3) y 2  272 x 2  1 The initial solution of pellian equation is

Key Words: Binary, Quadratic, Pyramidal numbers, integral solutions

y 2  272 x 2  16

( x0 , y0 ) is,

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(7)

n by n  1 in (7),we get

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7.528 yn3  4 xn1  8708xn3  0

272 xn3  264 272 f n1  4354 g n1  264 272 (33 f n  2 272 g n )  4354(33g n  2 272 g n ) 272 xn3  17420 272 f n  287298g n

8.2 yn 2  1088xn1  66 yn1  0

(8)

For simplicity and clear understanding the other choices of

9.2 yn3  71808xn1  4354 yn1  0

integer solutions are presented below:

10. 66 yn3  1088xn1  2 yn 2  0

272 yn1  66 272 f n  1088g n

(9)

272 yn 2  4354 272 f n  71808g n

(10)

272 yn3  287298 272 f n  4738240 g n

(11)

11. 8 yn1  8708xn 2  132 xn3  0 12. 8 yn 2  132 xn 2  4 xn3  0 13. 8 yn3  4 xn 2  132 xn3  0 14. 132 yn 2  2176xn 2  4 yn1  0

These are representing the non-zero distinct integer

15.132 yn3  143616 xn 2  132 yn1  0

solutions of (1)

16.4 yn3  2176 xn 2  132 yn 2  0

Some few numerical examples are given in the following table:

17.8708 yn 2  2176 xn3  132 yn1  0

Table 1: Numerical examples

n

xn 1

yn 1

-1 0 1 2 3 4

8 528 34840 2298912 151693352 10009462320

132 8708 574596 37914628 2501790852 165080281604

18.8708 yn3  143616 xn3  4 yn1  0 19.2176 xn3  4 yn 2  132 yn3  0

20. 2176 yn1  143616 yn 2  2176 yn3  0 21.Each of the following expression represents a cubical integer: 1 66 x3n 4  4354 x3n3  198 xn 2    8  13062 xn1 

The recurrence satisfied by the values of xn 1 and yn 1 are respectively xn3  66 xn 2  xn1  0 n  1,0,1,...... , yn3  66 yn 2  yn1  0 n  1,0,1,..... ,



1 66 x3n5  287298 x3n3  198 xn1    528  861894 xn1 



1 66 y3n3  1088x3n 3  198 y n1    4   3264 xn1 



1  y3n 4  1088 x3n3  3 y n 2    2  3264 xn1 



1 66 y3n5  4738240 x3n3  198 y n3    8708  14214720 xn1 

3. 4 yn 2  2 xn1  66 xn 2  0



1 4354 x3n5  287298 x3n 4  13062 xn1    8  861894 xn 2 

4. 4 yn3  66 xn1  4354 xn 2  0



1 4354 y3n3  1088 x3n 4  13062 y n1    132  3264 xn 2 



1 4354 y3n 4  71808 x3n 4  13062 y n 2    4  215424 xn 2 

A few interesting relations among the solutions are presented below: 1. 2 xn3  132 xn 2  2 xn1  0 2. 4 yn1  66 xn1  2 xn 2  0

5. 528 yn1  8708xn1  4 xn3  0

6.528 yn 2  132xn1  132xn3  0

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1 4354 y3n5  4738240 x3n 4  13062 y n1    132  14214720 xn 2 



1 287298 y3n 3  1088x3n 5     8708  861894 y n1  3264 xn 3  1 287298 y3n 4  71808 x3n5  861894 y n 2     132  215424 xn3 

1 287298 y3n5  4738240 x3n5  861894 y n3     4  14214720 xn3  1 718088 y3n3  1088 y3n 4  215424 y n1    2176  3264 y n 2 



 1 287298 y 4n5  71808 x4n 6   132  1149192 y 2n3  287232 x2n 4  798



 1 287298 y 4n 6  4738240 x4n 6   4  1149192 y 2n 4  18952960 x2n 4  24



 1 71808 y 4n 4  1088 y 4n5   2176  287232 y 2n 2  4352 y 2n3  13056



 4738240 y 4n 4  1088 y 4n 6  1    18952960 y 2 n 2  4352 y 2 n 4  143616    861696 



 4738240 y 4 n 5  71808 y 4n 6  1    18952960 y 2 n 3  287232 y 2 n 4   2176   13056 

23. Each of the following expression represents quintic integer:

1 4738240 y3n 3  1088 y3n5     143616  14214720 y n1  3264 y n3  1 4738240 y3n 4  71808 y3n5    2176  14214720 y n 2  215424 y n3 





22. Each of the following expression represents biquadratic integer:



 1 66 x5n 6  4354 x5n5  330 x3n 4   8  21770 x3n3  660 xn 2  43540 xn1 



66 y5n 7  287298 x5n 5  330 x3n 5  1    1436490 x3n 3  2872980 xn1   528  660 xn 3 



1 66 x4n5  4354 x4n 4  264 x2n3    8  17416 x2n 2  48 



1 66 y5n5  1088 x5n5  330 y3n3    4  5440 x3n3  10880 xn1  660 y n1 



1 66 y 4n 6  287298 x4n 4  264 x2n 4    528  1149192 x2n 2  1056 



1  y5n 6  1088 x5n5  5440 x3n 3    2  5 y3n 4  10880 xn1  10 y n 2 



1 66 y 4n 4  1088 x4n 4  264 y 2n 2    4  4352 x2n 2  24 



1  y 4n5  1088 x4n 4  4 y 2n3    2  4352 x2n 2  12 









66 y 4n 6  4738240 x4n 4  264 y 2n 4   1    8708 18952960 x2n 2  52248   1 4354 x4n5  287298 x4n5  17416 x2n 4    8  1149192 x2n3  48 





1 4354 y 4n 4  1088 x4n5  17416 y 2n 2    132  4352 x2n3  792 



1 4354 y 4n5  71808 x4n5  17416 y 2n3    4  287232 x2n3  54 



1 4354 y 4n 6  4738240 x4n5  17416 y 2n 4    132  18952960 x2n3  792 



1 287298 y 4n 4  1088 x4n 6  1149192 y 2n 2    8708  4352 x2n 4  52248 

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

|



66 x5n 7  4738240 x5n 5  330 y3n 5  1    23691200 x3n 3  47382400 xn1  8708   660 y n 3  4354 y5n 7  287298 x5n 6  1436490 x3n 4  1   21770 x3n 5  2872980 xn 2  8  43540 xn 3   4354 y5n 5  1088 x5n 6  5440 x3n 4  1    21770 y3n 3  1088 xn 2  132    43540 y n1   4354 y5n 6  71808 x5n 6  359040 x3n 4  1   21770 y3n 4  718080 xn 2  4   43540 y n 2  4738240 y5n 5  1088 y5n 7  1    5440 y3n 5  23691200 y3n 3  143616   10880 y n 3  47382400 y n1 

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











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 4354 y5n 7  4738240 x5n 6  1    23691200 x3n 4  21770 y3n 5  132    47382400 xn 2  43540 y n 3  287298 y5n 5  1088 x5n 7  5440 x3n 5  1    1436490 y3n 3  10880 xn 3  8708   2872980 y n1 

1 26112  430848 y2n 2  6528 y2n3  2176 1 1723392  28429440 y 2n 2     143616   6528 y 2n 4 





25. REMARKABLE OBSERVATIONS

 287298 y5n 6  71808 x5n 7  1    1436490 y3n 4  3590940 x3n 5  132    718080 xn 3  2872980 y n 2  287298 y5n 7  4738240 x5n 7  1   1436490 y3n 5  23691200 x3n 5  4  47382400 xn 3  2872980 y n 3 

I. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbolas which are presented in table 2 below: Table 2: Hyperbola

S.N O Hyperbola

71808 y5n 5  1088 y 5n 6  1    359040 y3n 3  5440 y3n 4  2176    1088 y n 2  718080 y n1   4738240 y5n 6  71808 y5n 7  1    23691200 y3n 4  359040 y3n 5  2176    47382400 y n 2  718080 y n 3 

1

          

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X n2  272Yn2  1115136

3

1 96  396 x2n3  26124x2n 2  8 1 6336  396 x2n 4  1723788x2n 2  528 1 48  396 y2n 2  6528x2n 2  4 1 24  6 y2n3  6528x2n 2  2 1 104496  396 y2n 4  28429440 x2n 2  8708 1 96  26124 x2n 4  1723788x2n3  8 1 1584  26124 y2n 2  6528x2n3  132 1 48  26124 y2n3  430848x2n3  4 1 1584  26124 y2n 4  28429440 x2n3  132 1 104496  1723788 y2n 2  6528x2n 4  8708 1 1584  1723788 y2n3  430848x2n 4  132 1 48  1723788 y2n 4  28429440 x2n 4  4

|

X n2  4352Yn2  256

24. Each of the following expression represents Nasty number: 

1 26112  28429440 y 2n3    2176  430848 y 2n 2 

X n2  272Yn2  64

 66 xn 2  4354 xn1 ,     66 xn1  xn 2   17420 xn1  4 xn3 ,     66 xn3  287298 xn1   66 y n1  1088 xn1 ,     66 xn1  4 y n1 

4

4356 X n2  272Yn2  y n 2  1088 x n1 ,     69696  4354 x n1  4 y n 2 

5

X n2  272Yn2  303317056

6

X n2  272Yn2  256

7

X n2  272Yn2  69696

8

X n2  272Yn2  64

9

X n2  272Yn2  69696

|

( X n ,Yn )

 66 y n 3  4738240 x n1 ,     287298 x n1  4 y n 3   4354 xn 3  287298 xn 2 ,     17420 n 2  264 xn 3 

 4354 y n1  1088 x n 2 ,     66 x n 2  264 y n1   4354 y n 2  71808 x n 2 ,     4354 x n 2  264 y n 2   4354 yn  3  4738240 xn  2 ,     287298 xn  2  264 yn  3 

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X n2  272Yn2  303317056

11

X n2  272Yn2  69696

12

X n2  272Yn2

13

X  272Y

2 n

 18939904

14

X n2  272Yn2  82502221824

15

 287298 y n1  1088 x n 3 ,     66 x n 3  17420 y n1 

8

 287298 y n 2  71808 x n 3 ,     4354 x n 3  17420 y n 2 

9

X n2  272Yn2

132 X n  272Yn2  34848

10

8708 X n  272Yn2  151658528

 71808 yn 1  1088 y n  2 ,     66 y n  2  4354 yn 1 

11

 4738240 yn 1  1088 yn  3 ,     66 yn  3  287298 yn 1 

12

132 X n  272Yn2  34848 4 X n  272Yn2  32

 4738240 y n 2  71808 y n 3 ,     4354 y n 3  287298 y n 2 

 18939904

4 X n  272Yn2  32

 287298 yn  3  4738240 xn  3 ,     287298 xn  3  17420 yn  3 

 64 2 n

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13

2176 X n  272 y n2  9469952

II. Employing linear combination among the solutions of (1), one may be generate integer solutions for other choices of parabolas which are presented in table 3 below:

14

Table 3: Parabola

S. NO

Parabola

 X n ,Yn 

1

X n  544Yn2

 16  66 x 2n 3  4354 x 2n 2 ,     66 x n1  x n 2 

 16

2

528 X n  272Yn2

X n  68Yn2 8

4

8712 X n  272Y

2 n

 34848

5

2177 X n  63Yn2  37914632

6

8 X n  272Yn2  128

7

132 X n  272Yn2  34848

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 17416  287298 y 2 n 2      1088 x 2n 4 ,   66 x  17420 y  n 3 n 1    264  287298 y 2 n 3      71808 x 2 n 4 ,   4354 x  17420 y  n 3 n 2    8  287298 y 2 n 4     4738240 x ,   2n 4  287298 x  17420 y  n 3 n 3  

 4352  71808 y n 2      1088 y 2 n 3 ,   66 y   4354 y n 2 n 1  

143616 X n  272Yn2  287232  4738240 y 2 n 2     41251110912   1088 y 2 n 4 ,   66 y  287298 y  n 3 n 1   2 2176 X n  272Yn  4352  4738240 y 2 n 3     9469952   71808 y 2 n 4 ,   4354 y  2 n  4  287298 y n  3  

Table 4: Straight line

 4  y 2n 3  1088 x 2n 2 ,     4354 x n1  4 y n 2   17416  66 y 2n 4     4738240 x ,   n 2  287298 x  4 y  n 1 n 3    16  4354 x 2 n 4      287298 x 2 n 3 ,   17420 x   264 x n 2 n 3  

 264  4354 y 2 n 2      1088 x 2 n 3 ,   66 x  n  2  264 y n 1  

Impact Factor value: 7.211

 264  4354 y 2n 4      4738240 x 2n 3 ,   287298 x  n  2  264 y n  3  

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

 1056  66 x 2 n 4      287298 x 2n 2 ,   17420 x  4 x  n 1 n 3    8  66 y 2n 2  1088 x 2n 2 ,     66 x n1  4 y n1 

 557568

3

15

 8  4354 y 2 n 3      71808 x 2 n 3 ,   4354 x  n  2  264 y n  2  

|

S.NO

Straight line

1

Y  66 X

 66 x n 2  4354 x n1 ,     66 x n 3  287298 x n1 

2

Y  2X

 66 y n1  1088 x n1 ,     66 x n 2  4354 x n1 

3

Y  4X

 y n 2  1088 x n1 ,     66 x n 2  4354 x n1 

4

YX

 4354 x n 3  287298 x n 2 ,     66 x n 2  4354 x n1 

5

Y  2X

 287298 y n 3  4738240 x n 3 ,     66 x n 2  4354 x n1 

X , Y 

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6

Y  272 X

 66 x n 2  4354 x n1 ,     71808 y n1  1088 y n 2 

7

Y  17952 X

 4354 x n 3  287298 x n 2 ,     4738240 y n1  1088 y n 3 

8

Y  272 X

 66 x n 2  4354 x n1 ,    4738240 y  71808 y n 2 n 3  

9

Y  132 X

 66 y n1  1088 x n1 ,     66 x n 3 ,287238 x n1 

10

Y  2X

 y n 2  1088 x n1 ,     66 y n1  1088 x n1 

11

Y  66 X

 4354 x n 3  287298 x n 2 ,     66 x n 3  287298 x n1 

12

Y  4X

 4354 y n1  1088 x n 2 ,     66 x n 3  287298 x n1 

13

YX

 4354 y n 3  4738240 x n 2 ,     4354 y n1  1088 x n 2 

14

Y  33 X

 287298 y n 3  4738240 x n 3 ,     287298 y n 2  71808 x n 3 

V. Employing the solutions in terms of special integers sequence namely, generalized Fibonacci sequence , GFn(k,s) and the generalized Lucas sequence GLn(k,s) are exhibited below:

xn1  4GLn1 (66,1)  264GFn1 (66,1) yn1  66GLn1 (66,1)  4352GFn1 (66,1) VI. Employing the solutions of (1),each of the following among the special polygonal,pyramidal,star numbers,pronic numbers ,centered pyramidal number is a congruent to under modulo 16.

2

i. ii.

5  12 p 5y     272 p x  t 3, x  s y 1  1    

 3 p 3y  2   t 3, y  2 

   

2

2

3 2    272 6 p x   prx 1      2

iii.

 4 pt y 3   4     272 6 p y 1   p3   t 3, 2 y 1     y 3 

iv.

 2 p y 1       272 4 pt x 3   p3   t 4, y 1   x 3   

v.

IV. Consider p  xn 1  yn 1 , q  xn 1

3 2  6 p 4y 1     272 3 p x   t 3, x 1   t 3, 2 y 1     

Note that p  q  0 , treat p, q as the generators of the Pythagorean triangle T ( X , Y , Z )

vi.

 4 p 5y   3    272 3 p x  2  t 3, y  2  ct 4, y 1    

vii.

 3 p 3y 2   5    272 p x t t   3, x  3, y  2 

15

2

 287238 y n1  1088 x n 3 ,     66 y n 3  4738240 x n1 

YX

2

where X  2 pq, Y  p  q , Z  p  q , p  q  0 2

2

2

2

2

2

2

2

It is odserved that T ( X , Y , Z ) is satisfied by the following relations:

   

   

2

2

2

X  136Y  135Z  16

5   36 p 3y  2      272 12 p x   s y 1  1   s x 1  1     

ii.

2A  xn 1 yn 1 P

2 5  6 p 2y 1     272 4 p x   ct 4, x1   t 3, 2 y 1     

iii.

4A X Y  Z P

iv.

3(Z  Y ) is a Nasty number

CONCLUSION

v.

4A  Y is written as the sum of two P squares

In this dissertation , we have presented infinitely many integer solutions of the binary quadratic Diophantine equation are rich in variety, one may search for the other choices of binary quadratic Diophantine equation and determine their integral solution along with suitable properties.

i.

viii.

2

2

ix.

2

x.

X

where A, P represents the area and perimeter of T ( X , Y , Z )

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Impact Factor value: 7.211

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2 5  4 p 5y     272 6 p x   ct 6, x 1   ct 4, y 1     

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Volume: 06 Issue: 03 | Mar 2019

p-ISSN: 2395-0072

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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 Publications,New York 1959.

[4]

Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation on the hyperbola x 2  19 y 2  3 ”, Scholars Journal of the Engineering and Technology 2014;2(2A);152-155.

[5]

Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation on the hyperbola y 2  26 x 2  1 , Bessel Math,Vol 4, Issue(1),2014;21-25.

[6]

On special families of Hyperbola, x 2  (4k 2  k ) y 2  a 2 , a  1 The International Journal of Science and Technology ,Vol 2,Issue(3),Pg.No.94-97, March 2014.

[7]

Dr.G.Sumathi “Observations on the hyperbola y 2  182 x 2  14 ”, Journal of Mathematics and Informatics,Vol 11, 73-81,2017

[8]

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

[9]

Dr.G.Sumathi “Observation on the Hyperbola y 2  312 x 2  1 ” International Journal of Mathematics Trends and Technology,Vol 50,Issue 4,31-34,Oct 2017

[10]

Dr.G.Sumathi “Observation on the Hyperbola 2 2 y  150 x  16 ” International Journal of recent Trends in Engineering and Research ,Vol 3,Issue 9,Pp198-206,Sep 2017

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