IRJET- On the Binary Quadratic Diophantine Equation Y2=272x2+16
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This document presents an analysis of the binary quadratic Diophantine equation y^2 = 272x^2 + 16. It finds the integral solutions to this equation and explores some properties among the solutions. Several recurrence relations satisfied by the solutions are provided. It is shown that certain expressions involving the solutions represent cubical and biquadratic integers. Pythagorean triangles corresponding to some of the integral solutions are also obtained.
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072
© 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1587
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.
---------------------------------------------------------------------***----------------------------------------------------------------------
Abstract – The binary quadratic equation
16272 22
xy is considered and a few interesting
properties among the solutions are presented .Employing
the integral solutions of the equation under
considerations a few patterns of Pythagorean triangle are
observed.
Key Words: Binary, Quadratic, Pyramidal numbers,integral
solutions
1. INTRODUCTION
The binary quadratic equation of the form 122
Dxy
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
16272 22
xy 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 Notations
nmt , : Polygonal number of rank n with size m
m
nP : Pyramidal number of rank n with size m
nPr : Pronic number of rank n
nS : Star number of rank n
nmCt , : Centered Pyramidal number of rank n with
size m
),( skGFn : Generalized Fibonacci sequence of rank n
),( skGLn : Generalized Lucas sequence of rank n
2. METHOD OF ANALYSIS
Consider the binary quadratic Diophantine equation is
16272 22
xy (1)
whose smallest positive integer solutions of ),( 00 yx is,
80 x , 1320 y (2)
To obtain the other solutions of (1),Consider pellian
equation is
1272 22
xy (3)
The initial solution of pellian equation is
2~
0 x , 33~
0 y
whose general solution nn yx ~,~ of (3) is given by,
nn gx
272
1~ , nn fy
2
1~
where,
11
)27233()272233(
nn
nf
11
)272233()272233(
nn
ng
Applying Brahmagupta lemma between 00, yx and nn yx ~,~
the other integer solution of (1) are given by,
nnn gfx
272
66
41 (4)
nnn gfy
20
40
91 (5)
Therefore (4) becomes
nnn gfx 662724272 1 (6)
Replace n by 1n in (6),we get
112 662724272 nnn gfx
)272233(66)272233(2724 nnnn fggf
nnn gfx 4354272264272 2 (7)
Replace n by 1n in (7),we get](https://image.slidesharecdn.com/irjet-v6i3299-190817091458/85/IRJET-On-the-Binary-Quadratic-Diophantine-Equation-Y2-272x2-16-1-320.jpg)
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IRJET- On the Binary Quadratic Diophantine Equation Y2=272x2+16
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1587 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. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract – The binary quadratic equation 16272 22 xy is considered and a few interesting properties among the solutions are presented .Employing the integral solutions of the equation under considerations a few patterns of Pythagorean triangle are observed. Key Words: Binary, Quadratic, Pyramidal numbers,integral solutions 1. INTRODUCTION The binary quadratic equation of the form 122 Dxy 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 16272 22 xy 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 Notations nmt , : Polygonal number of rank n with size m m nP : Pyramidal number of rank n with size m nPr : Pronic number of rank n nS : Star number of rank n nmCt , : Centered Pyramidal number of rank n with size m ),( skGFn : Generalized Fibonacci sequence of rank n ),( skGLn : Generalized Lucas sequence of rank n 2. METHOD OF ANALYSIS Consider the binary quadratic Diophantine equation is 16272 22 xy (1) whose smallest positive integer solutions of ),( 00 yx is, 80 x , 1320 y (2) To obtain the other solutions of (1),Consider pellian equation is 1272 22 xy (3) The initial solution of pellian equation is 2~ 0 x , 33~ 0 y whose general solution nn yx ~,~ of (3) is given by, nn gx 272 1~ , nn fy 2 1~ where, 11 )27233()272233( nn nf 11 )272233()272233( nn ng Applying Brahmagupta lemma between 00, yx and nn yx ~,~ the other integer solution of (1) are given by, nnn gfx 272 66 41 (4) nnn gfy 20 40 91 (5) Therefore (4) becomes nnn gfx 662724272 1 (6) Replace n by 1n in (6),we get 112 662724272 nnn gfx )272233(66)272233(2724 nnnn fggf nnn gfx 4354272264272 2 (7) Replace n by 1n in (7),we get
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1588 113 4354272264272 nnn gfx )272233(4354)272233(272264 nnnn gggf nnn gfx 28729827217420272 3 (8) For simplicity and clear understanding the other choices of integer solutions are presented below: nnn gfy 108827266272 1 (9) nnn gfy 718082724354272 2 (10) nnn gfy 4738240272287298272 3 (11) These are representing the non-zero distinct integer solutions of (1) Some few numerical examples are givenin thefollowing table: Table 1: Numerical examples The recurrence satisfied by the values of 1nx and 1ny are respectively 066 123 nnn xxx , ,......1,0,1n 066 123 nnn yyy , ,.....1,0,1n A few interesting relations among the solutions are presented below: 1. 021322 123 nnn xxx 2. 02664 211 nnn xxy 3. 06624 212 nnn xxy 4. 04354664 213 nnn xxy 5. 048708528 311 nnn xxy 0132132528.6 312 nnn xxy 087084528.7 313 nnn xxy 06610882.8 112 nnn yxy 04354718082.9 113 nnn yxy 10. 02108866 213 nnn yxy 11. 013287088 321 nnn xxy 12. 041328 322 nnn xxy 13. 013248 323 nnn xxy 14. 042176132 122 nnn yxy 0132143616132.15 123 nnn yxy 013221764.16 223 nnn yxy 013221768708.17 132 nnn yxy 041436168708.18 133 nnn yxy 013242176.19 323 nnn yyx 20. 021761436162176 321 nnn yyy 21.Each of the following expression representsacubical integer: 1 23343 13062 198435466 8 1 n nnn x xxx 1 13353 861894 19828729866 528 1 n nnn x xxx 1 13333 3264 198108866 4 1 n nnn x yxy 1 23343 3264 31088 2 1 n nnn x yxy 1 33353 14214720 198473824066 8708 1 n nnn x yxy 2 14353 861894 130622872984354 8 1 n nnn x xxx 2 14333 3264 1306210884354 132 1 n nnn x yxy 2 24343 215424 13062718084354 4 1 n nnn x yxy n 1nx 1ny -1 8 132 0 528 8708 1 34840 574596 2 2298912 37914628 3 151693352 2501790852 4 10009462320 165080281604
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1589 2 14353 14214720 1306247382404354 132 1 n nnn x yxy 31 5333 3264861894 1088287298 8708 1 nn nn xy xy 3 25343 215424 86189471808287298 132 1 n nnn x yxy 3 35353 14214720 8618944738240287298 4 1 n nnn x yxy 2 14333 3264 2154241088718088 2176 1 n nnn y yyy 31 5333 326414214720 10884738240 143616 1 nn nn yy yy 32 5343 21542414214720 718084738240 2176 1 nn nn yy yy 22. Each of the following expression represents bi- quadratic integer: 4817416 264435466 8 1 22 324454 n nnn x xxx 10561149192 26428729866 528 1 22 424464 n nnn x xxy 244352 264108866 4 1 22 224444 n nnn x yxy 124352 41088 2 1 22 324454 n nnn x yxy 5224818952960 264473824066 8708 1 22 424464 n nnn x yxy 481149192 174162872984354 8 1 32 425454 n nnn x xxx 7924352 1741610884354 132 1 32 225444 n nnn x yxy 54287232 17416718084354 4 1 32 325454 n nnn x yxy 79218952960 1741647382404354 132 1 32 425464 n nnn x yxy 522484352 11491921088287298 8708 1 42 226444 n nnn x yxy 7982872321149192 71808287298 132 1 4232 6454 nn nn xy xy 24189529601149192 4738240287298 4 1 4242 6464 nn nn xy xy 130564352287232 108871808 2176 1 3222 5444 nn nn yy yy 861696 435218952960 10884738240 143616 1 4222 6444 nn nn yy yy 13056 28723218952960 718084738240 2176 1 4232 6454 nn nn yy yy 23. Each of the following expression represents quintic integer: 1233 435565 4354066021770 330435466 8 1 nnn nnn xxx xxx 3 133 535575 660 28729801436490 33028729866 528 1 n nn nnn x xx xxy 1133 335555 660108805440 330108866 4 1 nnn nnn yxx yxy 2143 335565 10108805 54401088 2 1 nnn nnn yxy xxy 3 133 535575 660 4738240023691200 330473824066 8708 1 n nn nnn y xx yxx 3 253 436575 43540 287298021770 14364902872984354 8 1 n nn nnn x xx xxy 1 233 436555 43540 108821770 544010884354 132 1 n nn nnn y xy xxy 2 243 436565 43540 71808021770 359040718084354 4 1 n nn nnn y xy xxy 13 3353 7555 4738240010880 236912005440 10884738240 143616 1 nn nn nn yy yy yy
- 4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1590 32 5343 6575 4354047382400 2177023691200 47382404354 132 1 nn nn nn yx yx xy 1 333 537555 2872980 108801436490 54401088287298 8708 1 n nn nnn y xy xxy 23 5343 7565 2872980718080 35909401436490 71808287298 132 1 nn nn nn yx xy xy 33 5353 7575 287298047382400 236912001436490 4738240287298 4 1 nn nn nn yx xy xy 12 4333 6555 7180801088 5440359040 108871808 2176 1 nn nn nn yy yy yy 32 5343 7565 71808047382400 35904023691200 718084738240 2176 1 nn nn nn yy yy yy 24. Each of the following expression represents Nasty number: 2232 2612439696 8 1 nn xx 2242 17237883966336 528 1 nn xx 2222 652839648 4 1 nn xy 2232 6528624 2 1 nn xy 2242 28429440396104496 8708 1 nn xy 3242 17237882612496 8 1 nn xx 3222 6528261241584 132 1 nn xy 3232 4308482612448 4 1 nn xy 3242 28429440261241584 132 1 nn xy 4222 65281723788104496 8708 1 nn xy 4232 43084817237881584 132 1 nn xy 4242 28429440172378848 4 1 nn xy 3222 652843084826112 2176 1 nn yy 42 22 6528 284294401723392 143616 1 n n y y 22 32 430848 2842944026112 2176 1 n n y y 25. 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: Hyperbola S.N O Hyperbola ( nn YX , ) 1 256 4352 22 nn YX 21 12 66 ,435466 nn nn xx xx 2 1115136 272 22 nn YX 13 31 28729866 ,417420 nn nn xx xx 3 64 272 22 nn YX 11 11 466 ,108866 nn nn yx xy 4 69696 2724356 22 nn YX 21 12 44354 ,1088 nn nn yx xy 5 303317056 272 22 nn YX 31 13 4287298 ,473824066 nn nn yx xy 6 256 272 22 nn YX 32 23 26417420 ,2872984354 nn nn x xx 7 69696 272 22 nn YX 12 21 26466 ,10884354 nn nn yx xy 8 64 272 22 nn YX 22 22 2644354 ,718084354 nn nn yx xy 9 69696 272 22 nn YX 32 23 264287298 ,47382404354 nn nn yx xy
- 5. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1591 10 303317056 272 22 nn YX 13 31 1742066 ,1088287298 nn nn yx xy 11 69696 272 22 nn YX 23 32 174204354 ,71808287298 nn nn yx xy 12 64 272 22 nn YX 33 33 17420287298 ,4738240287298 nn nn yx xy 13 18939904 272 22 nn YX 12 21 435466 ,108871808 nn nn yy yy 14 48250222182 272 22 nn YX 13 31 28729866 ,10884738240 nn nn yy yy 15 18939904 272 22 nn YX 23 32 2872984354 ,718084738240 nn nn yy yy II. Employing linear combination amongthesolutionsof(1), one may be generate integer solutions for other choices of parabolas which are presented in table 3 below: Table 3: Parabola S. NO Parabola nn YX , 1 16 544 2 nn YX 21 2232 66 ,43546616 nn nn xx xx 2 557568 272528 2 nn YX 31 22 42 417420 ,287298 661056 nn n n xx x x 3 8 68 2 nn YX 11 2222 466 ,1088668 nn nn yx xy 4 34848 2728712 2 nn YX 21 2232 44354 ,10884 nn nn yx xy 5 37914632 632177 2 nn YX 31 2 42 4287298 ,4738240 6617416 nn n n yx x y 6 128 2728 2 nn YX 32 32 42 26417420 ,287298 435416 nn n n xx x x 7 34848 272132 2 nn YX 12 32 22 26466 ,1088 4354264 nn n n yx x y 8 32 2724 2 nn YX 22 32 32 2644354 ,71808 43548 nn n n yx x y 9 34848 272132 2 nn YX 32 32 42 264287298 ,4738240 4354264 nn n n yx x y 10 151658528 2728708 2 nn YX 13 42 22 1742066 ,1088 28729817416 nn n n yx x y 11 34848 272132 2 nn YX 23 42 32 174204354 ,71808 287298264 nn n n yx x y 12 32 2724 2 nn YX 33 42 42 17420287298 ,4738240 2872988 nn n n yx x y 13 9469952 2722176 2 nn yX 12 32 2 435466 ,1088 718084352 nn n n yy y y 14 24125111091 272143616 2 nn YX 13 42 22 28729866 ,1088 4738240287232 nn n n yy y y 15 9469952 2722176 2 nn YX 342 42 32 2872984354 ,71808 47382404352 nn n n yy y y 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: Table 4: Straight line S.NO Straight line YX, 1 XY 66 13 12 28729866 ,435466 nn nn xx xx 2 XY 2 12 11 435466 ,108866 nn nn xx xy 3 XY 4 12 12 435466 ,1088 nn nn xx xy 4 XY 12 23 435466 ,2872984354 nn nn xx xx 5 XY 2 12 33 435466 ,4738240287298 nn nn xx xy
- 6. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1592 6 XY 272 21 12 108871808 ,435466 nn nn yy xx 7 XY 17952 31 23 10884738240 ,2872984354 nn nn yy xx 8 XY 272 32 12 718084738240 ,435466 nn nn yy xx 9 XY 132 13 11 287238,66 ,108866 nn nn xx xy 10 XY 2 11 12 108866 ,1088 nn nn xy xy 11 XY 66 13 23 28729866 ,2872984354 nn nn xx xx 12 XY 4 13 21 28729866 ,10884354 nn nn xx xy 13 XY 21 23 10884354 ,47382404354 nn nn xy xy 14 XY 33 32 33 71808287298 ,4738240287298 nn nn xy xy 15 XY 13 31 473824066 ,1088287238 nn nn xy xy IV. Consider 11 nn yxp , 1 nxq Note that 0 qp , treat qp, asthegeneratorsofthe Pythagorean triangle ),,( ZYXT where 0,,,2 2222 qpqpZqpYpqX It is odserved that ),,( ZYXT is satisfied by the following relations: i. 16135136 ZYX ii. 11 2 nn yx P A iii. ZY P A X 4 iv. )(3 YZ is a Nasty number v. Y P A X 4 is written as the sum of two squares where PA, represents the area and perimeter of ),,( ZYXT 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: )1,66(264)1,66(4 111 nnn GFGLx )1,66(4352)1,66(66 111 nnn GFGLy VI. Employing the solutions of (1),each of the following among the special polygonal,pyramidal,starnumbers,pronic numbers ,centered pyramidal number is a congruent to under modulo 16. i. 2 ,3 5 2 1 5 272 1 12 x x y y t p s p ii. 2 1 3 2 2,3 3 2 6 272 3 x x y y pr p t p iii. 2 12,3 4 1 2 3 3 3 6 272 4 y y y y t p p pt iv. 2 3 3 3 2 1,4 1 4 272 2 x x y y p pt t p v. 2 1,3 3 2 12,3 4 1 3 272 6 x x y y t p t p vi. 2 2,3 3 2 2 1,4 5 3 272 4 y x y y t p ct p vii. 2 ,3 5 2 2,3 3 2 272 3 x x y y t p t p viii. 2 1 5 2 1 3 2 1 12 272 1 36 y x x y s p s p ix. 2 1,4 5 2 12,3 2 1 4 272 6 x x y y ct p t p x. 2 1,6 5 2 1,4 5 6 272 4 x x y y ct p ct p CONCLUSION 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.
- 7. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1593 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,ThetheoryofNumbersandDiophantine Analysis,Dover Publications,New York 1959. [4] Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S,“Observation on the hyperbola 319 22 yx ”, Scholars Journal of the Engineering and Technology 2014;2(2A);152-155. [5] Gopalan.M.A, Sumathi.G,Vidhyalakshmi.S,“Observation on the hyperbola 126 22 xy , Bessel Math,Vol 4, Issue(1),2014;21-25. [6] On special families of Hyperbola, 1,)4( 2222 aaykkx TheInternational Journal of Science and Technology ,Vol 2,Issue(3),Pg.No.94-97, March 2014. [7] Dr.G.Sumathi “Observations on the hyperbola 14182 22 xy ”, Journal of Mathematics and Informatics,Vol 11, 73-81,2017 [8] Dr.G.Sumathi “Observations on the pell equation 414 22 xy ”, International Journal of Creative Research Thoughts,Vol 6,Issue 1,Pp1074-1084,March 2018 [9] Dr.G.Sumathi “Observation on the Hyperbola 1312 22 xy ” International Journal of Mathematics Trends and Technology,Vol 50,Issue 4,31-34,Oct 2017 [10] Dr.G.Sumathi “Observation on the Hyperbola 16150 22 xy ” International Journal of recent Trends in Engineering and Research ,Vol 3,Issue 9,Pp198-206,Sep 2017