IRJET- On the Positive Pell Equation Y2=35x2+29

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On the Positive Pell Equation y 2  35 x 2  29 M.A. Gopalan1, T. Mahalakshmi2, K. sevvanthi3 1Professor,

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

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

3PG Scholor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.

----------------------------------------------------------------------***--------------------------------------------------------------------2Assistant

Abstract – The binary quadratic Diophantine equation represented by the positive Pellian

y 2  35x 2  29 is analyzed for its

non-zero distinct integer solutions. A few interesting relations among the solutions are given. Employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas.

Key Words: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions 1. INTRODUCTION The binary quadratic equation of the form y 2  Dx 2  1 where D is non-square positive integer, has been selected by various mathematicians for its non-trivial integer solutions when D takes different integral values [1-4]. For an extensive review of various problems, one may refer [5-17]. In this communication, yet another an interesting equation given by y 2  35x 2  29 is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presented.

2. METHOD OF ANALYSIS The Positive pell equation representing hyperbola under consideration is

y 2  35x 2  29

(1)

The smallest positive integer solutions of (1) are

x0  1 , y0  8 To obtain the other solutions of (1), consider the Pell equation

y 2  35x 2  1

(2)

whose initial solution is

~ x0  1, ~ y0  6 xn , ~ yn ) of (2) is given by The general Solution ( ~ ~ xn 

1

1 gn , ~ yn  f n 2 2 35

where

f n  (6  35 ) n1  (6  35 ) n1

g n  (6  35 ) n1  (6  35 ) n1 , n  1, 0 ,1,...

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

xn1 

1 4 fn  gn 2 35

yn1  4 f n 

35 gn 2

The recurrence relations satisfied by the solutions x and y are given by

xn3  12 xn 2  xn1  0 yn3  12 yn 2  yn1  0 Some numerical examples of xn and y n satisfying (1) are given in the Table: 1 below: Table: 1 Examples

n

xn

yn

0

1

8

1

14

83

2

167

988

3

1990

11773

4

23713

140288

5

282566

1671683

6

3367079

19919908

From the above table, we observe some interesting relations among the solutions which are presented below: 

Both xn and y n values are alternatively odd and even.

2.1. Relations among the solutions are given below: 

xn 2  yn1  6 xn1



xn3  12 yn1  71xn1



y n 2  6 y n1  35xn1



yn3  71yn1  420 xn1



yn 2  6 xn 2  xn1



yn3  71xn 2  6 xn1

 12 xn 2  xn3  xn1

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

xn3  2 yn 2  xn1



6 yn3  71yn 2  35xn1



71xn3  12 yn3  xn1



6 xn3  yn1  71xn 2



6 yn1  71yn 2  35xn3



6 xn 2  xn3  yn 2

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

yn1  71yn3  420 xn3



xn 2  6 xn3  yn3



yn2  6 yn3  35xn3



6 yn2  yn1  35xn2



yn3  yn1  70 xn 2



yn3  12 yn 2  yn1



yn3  6 yn 2  35xn2

2.2. Each of the following expression is a nasty number:               

6 58  16 y2n2  70 x2n2  29 6 58  16 x2n3  166 x2n2  29 1 348  8x2n4  988x2n2  29 1 348  16 y2n3  980 x2n2  29 6 4118  16 y2n4  11690 x2n2  2059 1 348  166 y2n2  70 x2n3  29 6 4118  1976 y2n2  70x2n4  2059 6 58  28 y2n2  2 y2n3  29 1 348  167 y2n2  y2n4  29 6 58  166 x2n 4  1976 x2n3  29 6 58  166 y2n3  980 x2n3  29 1 348  166 y2n4  11690x2n3  29 1 348  1976 y2n3  980 x2n4  29 6 58  1976 y2n4  11690x2n4  29 6 58  334 y2n3  28 y2n4  29

2.3. Each of the following expressions is a cubical integer: 1 16 y3n3  70 x3n3  48 yn1  210 xn1   29 1  16 x3n 4  166 x3n3  48 xn 2  498 xn1  29

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1 4 x3n5  494 x3n3  12 xn3  1482 xn1  87 1  8 y3n 4  490 x3n3  24 yn 2  1470 xn1  87 1 16 y3n5  11690 x3n3  48 yn3  35070 xn1   2059 1  83 y3n3  35 x3n 4  249 yn1  105 xn 2  87 1 1976 y3n3  70 x3n5  5928 yn1  210 xn3   2059 1  28 y3n3  2 y3n 4  84 yn1  6 yn 2  29 1 167 y3n3  y3n5  501yn1  3 yn3   174 1 166 x3n5  1976 x3n4  498xn3  5928xn2   29 1 166 y3n4  980 x3n4  498 yn2  2940 xn2   29 1  83 y3n5  5845 x3n 4  249 yn3  17535 xn 2  87 1  988 y3n 4\  490 x3n5  2964 yn 2  1470 xn3  87 1 1976 y3n5\  11690 x3n5  5928 yn3  35070 xn3   29 1  334 y3n 4\  28 y3n5  1002 yn 2  84 yn3  29 

2.4. Each of the following expressions is a biquadratic integer: 1 16 y4n4  70 x4n4  64 y2n2  280 x2n2  174  29 1  16 x4 n5  166 x4 n 4  64 x2 n3 664 x2 n 2  174 29 1  4 x4 n6  494 x4 n 4  16 x2n 4  1976 x2 n 2  522 87 1  8 y4 n5  490 x4 n 4  32 y2 n3  1951x2 n 2  522 87 1 16 y4n6  11690 x4n4  64 y2n4  46760 x2n2  12354  2059 1  83 y4 n 4  35x4 n5  332 y2 n 2  140 x2n3  522 87 1 1976 y4n4  70x4n6  7904 y2n2  280x2n4  12354  2059 1  28 y4 n 4  2 y4 n5  112 y2 n 2  8 y2 n3  174 29 1 167 y4n4  y4n6  668 y2n2  4 y2n4  1044  174

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1 166 x4n6  1976 x4n5  664 x2n4  7904 x2n3  174 29 1 166 y4n5  980 x4n5  664 y2n3  3920 x2n3  174  29 1  83 y4 n 6  5845x4 n5  332 y2 n 4  23380 x2n3  522 87 1  988 y4 n5  490 x4 n6  3952 y2 n3  1960 x2n 4  522 87 1 1976 y4n6  11690x4n6  7904 y2n4  46760x2n4  174  29 1  334 y4 n5  28 y4 n 6  1336 y2 n3  112 y2 n 4  174 29



2.5. Each of the following expression is a quintic integer: 1 16 y5n5  70 x5n5  80 y3n3  350 x3n3  160 yn1  700xn1  29 1 16 x5n6  166 x5n5  80 x3n4  830 x3n3  160 xn2  1660 xn1   29 1  4 x5n 7  494 x5n5  20 x3n5  2470 x3n3  40 xn3  4940 xn1  87 1  8 y5n 6  490 x5n5  40 y3n 4  2450 x3n3  80 yn 2  4900 xn1  87 1 16 y5n7  11690 x5n5  80 y3n5  58450 x3n3  160 yn3  116900xn1   2059 1  83 y5n5  35x5n 6  415 y3n3  175x3n 4  830 yn1  350 xn 2  87 1 1976 y5n5  70 x5n7  9880 y3n3  350 y3n5  19760 yn1  700xn3   2059 1  28 y5n5  2 y5n6  140 y3n3  10 y3n 4  280 yn1  20 yn 2  29 1 167 y5n5  y5n7  835 y3n3  5 y3n5  1670 yn1  10 yn3   174 1  166 x5n 7  1976 x5n 6  830 x3n5  9880 x3n 4  1660 xn3  19760 xn 2  29 1  166 y5n 6  980 x5n 6  830 y3n 4  4900 x3n 4  1660 yn 2  9880 xn 2  29 1  83 y5n 7  5840 x5n 6  415 y3n5  29220 x3n 4  830 yn3  58450 xn 2  87 1  988 y5n 6  490 x5n 7  4940 y3n 4  2450 x3n5  9880 yn 2  4900 xn3  87



1 1976 y5n7  11690x5n7  9880 y3n5  58450x3n5  19760 yn3  116900xn3  29 1  334 y5n 6  28 y5n 7  1670 y3n 4  140 y3n5  3340 yn 2  280 yn3  29



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REMARKABLE OBSERVATIONS 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 Hyperbolas

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S.NO

Hyperbolas

(Y,X)

1

Y 2  35 X 2  117740

 16 yn1  70 xn1 ,     16 35 x  2 35 y  n1 n1  

2

Y 2  35 X 2  117740

16 xn 2  166 xn1 ,     28 35 x  2 35 x  n1 n 2  

3

Y 2  35 X 2  16954560

 16 xn3  1976 xn1 ,     334 35 x  2 35 x  n1 n 3  

4

Y 2  35 X 2  4238640

 16 yn 2  980 xn1 ,     166 35 x  2 35 y  n1 n 2  

5

Y 2  35 X 2  593527340

 16 yn3  11690 xn1 ,     1976 35 x  2 35 y  n1 n 3  

6

Y 2  35 X 2  4238640

 166 yn1  70 xn 2 ,     16 35 x  28 35 y  n 2 n1  

7

Y 2  35 X 2  593527340

1976 yn1  70 xn3 ,    16 35 x  334 35 y  n 3 n1  

8

35Y 2  X 2  4120900

 28 yn1  2 yn 2 ,     16 35 y  166 35 y  n 2 n1  

9

35Y 2  X 2  593409600

 334 yn1  2 yn3 ,     16 35 y  1976 35 y  n  3 n  1  

10

Y 2  35 X 2  117740

 166 xn3  1976 xn 2 ,     334 35 x  28 35 x  n 2 n 3  

11

Y 2  35 X 2  117740

 166 yn 2  980 xn 2 ,     166 35 x  28 35 y  n 2 n 2  

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Y 2  35 X 2  4238640

 166 yn3  11690 xn 2 ,     1976 35 x  28 35 y  n 2 n 3  

13

Y 2  35 X 2  4238640

 1976 yn 2  980 xn3 ,     166 35 x  334 35 y  n 3 n 2  

14

Y 2  35 X 2  117740

 1976 yn3  11690 xn3 ,     1976 35 x  334 35 y  n 3 n 3  

15

Y 2  35 X 2  144231500

 11690 yn 2  980 yn3 ,     166 35 y  1976 35 y  n 3 n 2  

Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabolas which are presented in Table:3 below: Table: 3 Parabolas

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S.NO

Parabolas

(Y,X)

1

29Y  X 2  3364

 16 y2 n 2  70 x2 n 2  58,     16 35 x  2 35 y  n1 n1  

2

29Y  X 2  3364

 16 x2 n3  166 x2 n 2  58,     28 35 x  2 35 x  n 1 n 2  

3

348Y  X 2  484416

 16 x2 n 4  1976 x2 n 2  696,     334 35 x  2 35 x  n 1 n 3  

4

174Y  X 2  121104

 16 y2 n3  980 x2 n 2  348,     166 35 x  2 35 y  n1 n 2  

5

2059Y  X 2  16957924

16 y2 n 4  11690 x2 n 2  4118,    1976 35 x  2 35 y  n1 n 3  

6

174Y  X 2  121104

166 y2 n 2  70 x2 n3  348,    16 35 x  28 35 y  n 2 n1  

7

2059Y  X 2  16957924

 1976 y2 n 2  70 x2 n 4  4118,     16 35 x  334 35 y  n 3 n1  

8

35525Y  X 2  4120900

 28 y2 n 2  2 y2 n3  58,     16 35 y  166 35 y  n 2 n1  

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 334 y2 n 2  2 y2 n 4  696,    16 35 y  1976 35 y  n 3 n1  

426300Y  X 2  593409600

10

29Y  X 2  3364

 166 x2 n 4  1976 x2 n3  58,     334 35 x  28 35 x  n 2 n 3  

11

29Y  X 2  3364

 166 y2 n3  980 x2 n3  58,     166 35 x  28 35 y  n 2 n 2  

12

166 y2 n 4  11690 x2 n3  348,    1976 35 x  28 35 y  n 2 n 3  

174Y  X 2  121104

13

174Y  X 2  121104

 1976 y2 n3  980 x2 n 4  348,     166 35 x  334 35 y  n 3 n 2  

14

29Y  X 2  3364

1976 y2 n 4  11690 x2 n 4  58,    1976 35 x  334 35 y  n 3 n 3  

15

1015Y  X 2  4120900

11690 y2 n3  980 y2 n 4  2030,    166 35 y  1976 35 y  n 3 n 2  

3. CONCLUSION In this paper, we have presented infinitely many integer solutions for the positive Pell equation y 2  35x 2  29 . As the binary quadratic Diophantine equations are rich in variety, one may search for the other choices of Pell equations and determine their integer solutions along with suitable properties.

REFERENCES [1] L.E. Dickson, History of Theory of numbers, Chelsea Publishing Company, volume-II, (1952) New York.

[2] L.J. Mordel, Diophantine Equations, Academic Press, (1969) New York.

[3] S.J. Telang, Number Theory, Tata McGraw Hill Publishing Company Limited, (2000) New Delhi.

[4] D.M. Burton, Elementary Number Theory, Tata McGraw Hill Publishing Company Limited, (2002) New Delhi.

[5] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, On the integral solutions of the binary quadratic Equation X 2  4( K 2  1)Y 2  4t , Bulletin of Mathematics and Statistics Research, 2(1)(2014), 42-46.

[6] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, On the binary quadratic Diophantine Equation x 2  3xy  y 2  33x  0 , International Journal of Scientific Engineering and Applied Science, 1(4)(2015), 222-225.

[7] M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha, Observations on the Hyperbola 10 y 2  3x 2  13, Archimedes J. Math., 3(1) (2013), 31-34.

[8] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Observations on the Hyperbola ax 2  (a  1) y 2  3a  1, Discovery, 4(10) (2013),22-24.

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[9] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Integral points on the Hyperbola x 2  4 xy  y 2  15x  0, Diophantus J.Maths,1(7)(2014), 338-340.

[10] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, on the integral solutions of the binary quadratic equation x 2  15y 2  11t , Scholars Journal of Engineering and Technology, 2(2A) (2014), 156-158.

[11] A.Kavitha and V.Kiruthika, On the positive Pell equation y 2  19 x 2  20, to appear in International Journal of Emerging Technologies in Engineering Research, 5(3) (2017), 71-74.

[12] A.Kavitha and P.Lavanya, On the positive Pell equation y 2  20 x 2  16, to appear in International Journal of Emerging Technologies in Engineering Research, 5(3) (2017), 75-79.

[13] K.Meena, M.A.Gopalan, T.Swetha, On the negative Pell equation y 2  40 x 2  4, International Journal of Emerging Technologies in Engineering Research, 5(1)(2017), 6-11.

[14] S.Ramya and A.Kavitha, On the positive Pell equation y 2  90 x 2  31, Journal of Mathematics and Informatics, (11)(2017), 111-117.

[15] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, On the Negative Pellian Equation y 2  20 x 2  31 , International Journal of Pure and Applied Mathematics, 119(14) (2018), 199-204.

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

[16] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, Observations on the Positive Pell Equation y 2  20 x 2  1 , International Journal of Pure and Applied Mathematics, 120(6) (2018), 11813-11825.

[17] M.A.Gopalan, A.Sathya, A. Nandhinidevi, Observation on the Positive Pell Equation y 2  15x 2  10 , International Journal for Research in Applied Science and Engineering Technology,vol 7,Issue II,Feb 2019,pp-

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