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IRJET- On Binary Quadratic Equation 2x2-3y2=-4

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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 973 ON BINARY QUADRATIC EQUATION 432 22  yx S. Vidhyalakshmi1, A. sathya2, A. Nandhinidevi3 1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. 2 Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. 3PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract- The binary quadratic Diophantine equation represented by the positive Pellian 432 22  yx 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 and Pythagorean triangle. Key Words: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions 1. INTRODUCTION The binary quadratic Diophantine equations of the form  0,,,22  NbaNbyax are rich in variety and have been analyzed by many mathematicians for their respective integer solutions for particular values of ba, and N . In this context, one may refer [1-14]. This communication concerns withtheproblemofobtaining non-zero distinct integer solutions to the binary quadratic equation given by 432 22  yx representinghyperbola.A few interesting relations among its solutions are presented. Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special Pythagorean triangle is constructed. 2. METHOD OF ANALYSIS: The Diophantine equation representingthebinary quadratic equation to be solved for its non-zero distinct integral solution is 432 22  yx (1) Consider the linear transformations TXyTXx 2,3  (2) From (1) and (2), we have 46 22  TX (3) whose smallest positive integer solution is 4,10 00  TX To obtain the other solutions of (3), consider the Pell equation 16 22  TX (4) Whose smallest positive integer solution is )2,5() ~ , ~ ( 00 TX the general solution of (4) is given by nnnn fXgT 2 1~ , 62 1~  Where 11 )625()625(   nn n f ....1,10 ,)625()625( 11    n g nn n Applying Brahmagupta lemma between )~,~( 00 yx and )~,~( nn yx the other integer solutions of (3) are given by nnn gfx 6 12 51  (5) nnn gfy 6 5 21  (6) From (2), (5) and (6) the values of x and y satisfying (1) are given by nnn gfx 6 27 111  nnn gfy 6 22 91  The recurrence relations satisfied by x and y are given by 010 123   nnn xxx 010 123   nnn yyy Some numerical examples of n x and n y satisfying (1) are given in the Table: 1 below,
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 974 Table: 1 Numerical Examples n n x n y 0 22 18 1 218 178 2 2158 1762 3 21362 17442 4 211462 172658 5 2093258 1709138 From the above table, we observe some interesting relations among the solutions which are presented below:  Both nx and ny values are even.  Relations among the solutions aregivenbelow:  010 123   nnn xxx  065 121   nnn yxx  065 121   nnn yxx  06495 321   nnn yxx  04960 133   nnn xxy  06049 113   nnn yxx  045 112   nnn xyy  012 213   nnn yxx  04495 123   nnn xyy  012585715807 133   nnn xyx  04049 131   nnn xyy  04956 231   nnn xxy  0211489213656 232   nnn xxy  056 233   nnn xxy  0539544 112   nnn yxx  045 212   nnn xyy  08 213   nnn xyy  065 223   nnn yxx  045 223   nnn xyy  04549 312   nnn xyy  04049 313   nnn xyy  045 323   nnn xyy  045 121   nnn xyy  010 123   nnn yyy  05494 321   nnn yyx  Each of the following expressions represents a nasty number:   2232 2672712   nn xx   2242 264327120 10 1   nn xx   2222 13216212   nn xy   2232 130816260 5 1   nn xy   2242 12948162588 49 1   nn xy   3242 264326712   nn xx   3222 132160260 5 1   nn xy   3232 1308160212   nn xy   3242 12948160260 5 1   nn xy   4222 13215858588 49 1   nn xy   4232 13081585860 5 1   nn xy   4242 129481585812   nn xy   3222 132130848 4 1   nn yy   4222 13212948480 40 1   nn yy   4232 13081294848 4 1   nn yy  Each of the following expressions represents a cubical integer:            1 23343 801 8126727 6 1 n nnn x xxx            1 33353 7929 81264327 60 1 n nnn x xxx            1 13333 66 812227 n nnn x yxy            1 23343 654 8121827 5 1 n nnn x yxy            1 33353 6474 81215827 49 1 n nnn x yxy            2 34353 7929 8012643267 6 1 n nnn x xxx
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 975            2 14333 66 80122267 5 1 n nnn x yxy            2 24343 654 801218267 n nnn x yxy            2 34353 6474 8012158267 5 1 n nnn x yxy            3 15333 66 7929222643 49 1 n nnn x yxy            3 25343 654 79292182643 5 1 n nnn x yxy            3 35353 6474 792921582643 n nnn x yxy            2 14333 66 65422218 4 1 n nnn y yyy            3 15333 66 6474222158 40 1 n nnn y yyy            3 25343 654 64742182158 4 1 n nnn y yyy  Each of the following expressions is a biquadratic integer:            361068 10826727 6 1 22 324454 n nnn x xxx            36010572 108264327 60 1 22 424464 n nnn x xxx            688 1082227 22 224444 n nnn x yxy            30872 10821827 5 1 22 324454 n nnn x yxy            2948632 108215827 49 1 22 424464 n nnn x yxy            3610572 10682643267 6 1 32 425464 n nnn x xxx            3088 106822267 5 1 32 225444 n nnn x yxy            6872 1068218267 32 325454 n nnn x yxy            308632 10682158267 5 1 32 425464 n nnn x yxy            29488 10572222643 49 1 42 226444 n nnn x yxy            30872 105722182643 5 1 42 326454 n nnn x yxy            68632 1057221582643 42 426464 n nnn x yxy            2488 87222218 4 1 32 225444 n nnn y yyy            24088 8632222158 40 1 42 226444 n nnn y yyy            24872 86322182158 4 1 42 326454 n nnn y yyy  Each of the following expression is a quintic integer:            1233 435565 26702701335 13526727 6 1 nnn nnn xxx xxx                  1 333 535575 26520 27013215 135264327 60 1 n nn nnn x xx xxx            1133 335555 220270110 1352227 nnn nnn xyx yxy            1233 435565 21802701090 13521827 5 1 nnn nnn xyx yxy                  1 333 535575 21580 27010790 135215827 49 1 n nn nnn x yx yxy            2343 536575 26430267013215 13352643267 6 1 nnn nnn xxx xxx            2143 336555 2202670110 133522267 5 1 nnn nnn xyx yxy            2243 436565 218026701090 1335218267 nnn nnn xyx yxy            2343 536575 21580267010790 13352158267 5 1 nnn nnn xyx yxy
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 976            3153 337555 22026430110 13215222643 49 1 nnn nnn xyx yxy            3253 437565 2180264301090 132152182643 5 1 nnn nnn xyx yxy            3353 537575 215802643010790 1321521582643 nnn nnn xyx yxy            2143 336555 2202180110 109022218 4 1 nnn nnn yyy yyy            3153 337555 22021580110 10790222158 40 1 nnn nnn yyy yyy            3253 437565 2180215801090 107902182158 4 1 nnn nnn yyy yyy 3. REMARKABLE OBSERVATIONS 3.1 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 S.No Hyperbola  YX , 1 1446 22  YX           21 12 11109 ,26727 nn nn xx xx 2 144006 22  YX           31 13 111079 ,264327 nn nn xx xx 3 46 22  YX           11 11 119 ,2227 nn nn yx xy 4 1006 22  YX           21 12 1189 ,21827 nn nn yx xy 5 96046 22  YX           31 13 11881 ,215827 nn nn yx xy 6 1446 22  YX           32 23 1091079 ,2643267 nn nn xx xx 7 1006 22  YX           12 21 1099 ,22267 nn nn yx xy 8 46 22  YX           22 22 10989 ,218267 nn nn yx xy 9 1006 22  YX           32 23 109881 ,2158267 nn nn yx xy 10 96046 22  YX           13 31 10799 ,222643 nn nn yx xy 11 1006 22  YX           23 32 107989 ,2182643 nn nn yx xy 12 46 22  YX           33 33 1079881 ,21582643 nn nn yx xy 13 646 22  YX           12 21 899 ,22218 nn nn yy yy 14 64006 22  YX           13 31 889 ,222158 nn nn yy yy 15 646 22  YX           23 32 88189 ,2182158 nn nn yy yy 3.2 Employing linear combination 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 S.No Parabola  YX , 1 122  XY           21 2232 11109 ,26727 nn nn xx xx 2 1200102  XY           31 2242 111079 ,264327 nn nn xx xx
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 977 3 26 2  XY           11 2222 119 ,2227 nn nn yx xy 4 5056 2  XY           21 2232 1189 ,21827 nn nn yx xy 5 4802496 2  XY           31 2242 11881 ,215827 nn nn yx xy 6 122  XY           32 3242 1091079 ,2643267 nn nn xx xx 7 5056 2  XY           12 3222 1099 ,22267 nn nn yx xy 8 26 2  XY           22 3232 10989 ,218267 nn nn yx xy 9 5056 2  XY           32 3242 109881 ,2158267 nn nn yx xy 10 4802496 2  XY           13 4222 10799 ,222643 nn nn yx xy 11 5056 2  XY           23 4232 107989 ,2182643 nn nn yx xy 12 26 2  XY           33 4242 1079881 ,21582643 nn nn yx xy 13 3246 2  XY           12 3222 899 ,22218 nn nn yy yy 14 3200406 2  XY           13 4222 889 ,222158 nn nn yy yy 15 3246 2  XY           23 4232 88189 ,2182158 nn nn yy yy 3.3 Let qp, be two non-zero distinct integers such that 0 qp . Treat qp, as the generators of the Pythagorean triangle   ,,T where 0,,,2 2222  qpqpqppq  . Taking 111 ,   nnn xqyxp , itisobservedthat   ,,T is satisfied by the following relations:  423      P A 4  11 2   nn yx P A where A, P represent the area and perimeter of the Pythagorean triangle. 4. CONCLUSION In this paper, we have presented infinitely many integer solutions for the positive Pell equation 432 22  yx . 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] R.D. Carmichael,The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1950). [2] L.E. Disckson, History of Theory of Numbers, vol II, Chelsea publishing Co., New York (1952). [3] L.J. Mordell, Diophantine Equations, Academic press, London (1969). [4] M.A. Gopalan and R. Anbuselvi, Integral solutions of   1314 22  axaay , Acta Ciencia Indica, XXXIV(1) (2008) 291-295. [5] M.A. Gopalan,et al., Integral points on the hyperbola     0,,2142 222 kakkaayxa  ,IndianJournalof Science, 1(2) (2012) 125-126. [6] M.A. Gopalan, S. Devibala and R. Vidhylakshmi, Integral points on the hyperbola 532 22  yx ,AmearicanJornal of Applied Mathematics and Mathematical Sciences, 1(2012) 1-4. [7] S. Vidhyalakshmi,et al., Observations on the hyperbola   131 22  ayaax , Discovery, 4(10) (2013) 22-24. [8] M.A. gopalan,et al., Integral points on the hyperbola     0,,2142 222 kakkaayxa  , Indian Journal of science, 1(2) (2012) 125-126. [9] M.A. Gopalan and S. Vidhyalakshmi and A. Kavitha, on the integer solutions of binary quadratic equation,   0,,414 222  tkykx t , BOMSR, 2(2014) 42-46. [10] T.R. Usha Rani and K. Ambika, Observation on the Non- Homogeneous Binary Quadratic Diophantine Equation 565 22  yx , Journal of Mathematics and Informatics, Vol-10,Dec (2017), 67-74. [11] M.A. Gopalan and Sharadha Kumar, On the Hyperbola 2332 22  yx , Journal of Mathematics and Informatics, vol-10,Dec (2017),1-9.
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 978 [12] M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha, Observationson the Hyperbola ,13310 22  xy Archimedes J. Math., 3(1) (2013), 31-34. [13] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, On the integral solutions of the binary quadratic equation ,1115 22 t yx  Scholars Journal of Engineering and Technology, 2(2A) (2014), 156-158. [14] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, Observations on the Positive Pell Equation  120 22  xy , International Journal of Pure and Applied Mathematics, 120(6) (2018), 11813-11825.

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IRJET- On Binary Quadratic Equation 2x2-3y2=-4

  • 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 973 ON BINARY QUADRATIC EQUATION 432 22  yx S. Vidhyalakshmi1, A. sathya2, A. Nandhinidevi3 1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. 2 Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. 3PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract- The binary quadratic Diophantine equation represented by the positive Pellian 432 22  yx 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 and Pythagorean triangle. Key Words: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions 1. INTRODUCTION The binary quadratic Diophantine equations of the form  0,,,22  NbaNbyax are rich in variety and have been analyzed by many mathematicians for their respective integer solutions for particular values of ba, and N . In this context, one may refer [1-14]. This communication concerns withtheproblemofobtaining non-zero distinct integer solutions to the binary quadratic equation given by 432 22  yx representinghyperbola.A few interesting relations among its solutions are presented. Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special Pythagorean triangle is constructed. 2. METHOD OF ANALYSIS: The Diophantine equation representingthebinary quadratic equation to be solved for its non-zero distinct integral solution is 432 22  yx (1) Consider the linear transformations TXyTXx 2,3  (2) From (1) and (2), we have 46 22  TX (3) whose smallest positive integer solution is 4,10 00  TX To obtain the other solutions of (3), consider the Pell equation 16 22  TX (4) Whose smallest positive integer solution is )2,5() ~ , ~ ( 00 TX the general solution of (4) is given by nnnn fXgT 2 1~ , 62 1~  Where 11 )625()625(   nn n f ....1,10 ,)625()625( 11    n g nn n Applying Brahmagupta lemma between )~,~( 00 yx and )~,~( nn yx the other integer solutions of (3) are given by nnn gfx 6 12 51  (5) nnn gfy 6 5 21  (6) From (2), (5) and (6) the values of x and y satisfying (1) are given by nnn gfx 6 27 111  nnn gfy 6 22 91  The recurrence relations satisfied by x and y are given by 010 123   nnn xxx 010 123   nnn yyy Some numerical examples of n x and n y satisfying (1) are given in the Table: 1 below,
  • 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 974 Table: 1 Numerical Examples n n x n y 0 22 18 1 218 178 2 2158 1762 3 21362 17442 4 211462 172658 5 2093258 1709138 From the above table, we observe some interesting relations among the solutions which are presented below:  Both nx and ny values are even.  Relations among the solutions aregivenbelow:  010 123   nnn xxx  065 121   nnn yxx  065 121   nnn yxx  06495 321   nnn yxx  04960 133   nnn xxy  06049 113   nnn yxx  045 112   nnn xyy  012 213   nnn yxx  04495 123   nnn xyy  012585715807 133   nnn xyx  04049 131   nnn xyy  04956 231   nnn xxy  0211489213656 232   nnn xxy  056 233   nnn xxy  0539544 112   nnn yxx  045 212   nnn xyy  08 213   nnn xyy  065 223   nnn yxx  045 223   nnn xyy  04549 312   nnn xyy  04049 313   nnn xyy  045 323   nnn xyy  045 121   nnn xyy  010 123   nnn yyy  05494 321   nnn yyx  Each of the following expressions represents a nasty number:   2232 2672712   nn xx   2242 264327120 10 1   nn xx   2222 13216212   nn xy   2232 130816260 5 1   nn xy   2242 12948162588 49 1   nn xy   3242 264326712   nn xx   3222 132160260 5 1   nn xy   3232 1308160212   nn xy   3242 12948160260 5 1   nn xy   4222 13215858588 49 1   nn xy   4232 13081585860 5 1   nn xy   4242 129481585812   nn xy   3222 132130848 4 1   nn yy   4222 13212948480 40 1   nn yy   4232 13081294848 4 1   nn yy  Each of the following expressions represents a cubical integer:            1 23343 801 8126727 6 1 n nnn x xxx            1 33353 7929 81264327 60 1 n nnn x xxx            1 13333 66 812227 n nnn x yxy            1 23343 654 8121827 5 1 n nnn x yxy            1 33353 6474 81215827 49 1 n nnn x yxy            2 34353 7929 8012643267 6 1 n nnn x xxx
  • 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 975            2 14333 66 80122267 5 1 n nnn x yxy            2 24343 654 801218267 n nnn x yxy            2 34353 6474 8012158267 5 1 n nnn x yxy            3 15333 66 7929222643 49 1 n nnn x yxy            3 25343 654 79292182643 5 1 n nnn x yxy            3 35353 6474 792921582643 n nnn x yxy            2 14333 66 65422218 4 1 n nnn y yyy            3 15333 66 6474222158 40 1 n nnn y yyy            3 25343 654 64742182158 4 1 n nnn y yyy  Each of the following expressions is a biquadratic integer:            361068 10826727 6 1 22 324454 n nnn x xxx            36010572 108264327 60 1 22 424464 n nnn x xxx            688 1082227 22 224444 n nnn x yxy            30872 10821827 5 1 22 324454 n nnn x yxy            2948632 108215827 49 1 22 424464 n nnn x yxy            3610572 10682643267 6 1 32 425464 n nnn x xxx            3088 106822267 5 1 32 225444 n nnn x yxy            6872 1068218267 32 325454 n nnn x yxy            308632 10682158267 5 1 32 425464 n nnn x yxy            29488 10572222643 49 1 42 226444 n nnn x yxy            30872 105722182643 5 1 42 326454 n nnn x yxy            68632 1057221582643 42 426464 n nnn x yxy            2488 87222218 4 1 32 225444 n nnn y yyy            24088 8632222158 40 1 42 226444 n nnn y yyy            24872 86322182158 4 1 42 326454 n nnn y yyy  Each of the following expression is a quintic integer:            1233 435565 26702701335 13526727 6 1 nnn nnn xxx xxx                  1 333 535575 26520 27013215 135264327 60 1 n nn nnn x xx xxx            1133 335555 220270110 1352227 nnn nnn xyx yxy            1233 435565 21802701090 13521827 5 1 nnn nnn xyx yxy                  1 333 535575 21580 27010790 135215827 49 1 n nn nnn x yx yxy            2343 536575 26430267013215 13352643267 6 1 nnn nnn xxx xxx            2143 336555 2202670110 133522267 5 1 nnn nnn xyx yxy            2243 436565 218026701090 1335218267 nnn nnn xyx yxy            2343 536575 21580267010790 13352158267 5 1 nnn nnn xyx yxy
  • 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 976            3153 337555 22026430110 13215222643 49 1 nnn nnn xyx yxy            3253 437565 2180264301090 132152182643 5 1 nnn nnn xyx yxy            3353 537575 215802643010790 1321521582643 nnn nnn xyx yxy            2143 336555 2202180110 109022218 4 1 nnn nnn yyy yyy            3153 337555 22021580110 10790222158 40 1 nnn nnn yyy yyy            3253 437565 2180215801090 107902182158 4 1 nnn nnn yyy yyy 3. REMARKABLE OBSERVATIONS 3.1 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 S.No Hyperbola  YX , 1 1446 22  YX           21 12 11109 ,26727 nn nn xx xx 2 144006 22  YX           31 13 111079 ,264327 nn nn xx xx 3 46 22  YX           11 11 119 ,2227 nn nn yx xy 4 1006 22  YX           21 12 1189 ,21827 nn nn yx xy 5 96046 22  YX           31 13 11881 ,215827 nn nn yx xy 6 1446 22  YX           32 23 1091079 ,2643267 nn nn xx xx 7 1006 22  YX           12 21 1099 ,22267 nn nn yx xy 8 46 22  YX           22 22 10989 ,218267 nn nn yx xy 9 1006 22  YX           32 23 109881 ,2158267 nn nn yx xy 10 96046 22  YX           13 31 10799 ,222643 nn nn yx xy 11 1006 22  YX           23 32 107989 ,2182643 nn nn yx xy 12 46 22  YX           33 33 1079881 ,21582643 nn nn yx xy 13 646 22  YX           12 21 899 ,22218 nn nn yy yy 14 64006 22  YX           13 31 889 ,222158 nn nn yy yy 15 646 22  YX           23 32 88189 ,2182158 nn nn yy yy 3.2 Employing linear combination 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 S.No Parabola  YX , 1 122  XY           21 2232 11109 ,26727 nn nn xx xx 2 1200102  XY           31 2242 111079 ,264327 nn nn xx xx
  • 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 977 3 26 2  XY           11 2222 119 ,2227 nn nn yx xy 4 5056 2  XY           21 2232 1189 ,21827 nn nn yx xy 5 4802496 2  XY           31 2242 11881 ,215827 nn nn yx xy 6 122  XY           32 3242 1091079 ,2643267 nn nn xx xx 7 5056 2  XY           12 3222 1099 ,22267 nn nn yx xy 8 26 2  XY           22 3232 10989 ,218267 nn nn yx xy 9 5056 2  XY           32 3242 109881 ,2158267 nn nn yx xy 10 4802496 2  XY           13 4222 10799 ,222643 nn nn yx xy 11 5056 2  XY           23 4232 107989 ,2182643 nn nn yx xy 12 26 2  XY           33 4242 1079881 ,21582643 nn nn yx xy 13 3246 2  XY           12 3222 899 ,22218 nn nn yy yy 14 3200406 2  XY           13 4222 889 ,222158 nn nn yy yy 15 3246 2  XY           23 4232 88189 ,2182158 nn nn yy yy 3.3 Let qp, be two non-zero distinct integers such that 0 qp . Treat qp, as the generators of the Pythagorean triangle   ,,T where 0,,,2 2222  qpqpqppq  . Taking 111 ,   nnn xqyxp , itisobservedthat   ,,T is satisfied by the following relations:  423      P A 4  11 2   nn yx P A where A, P represent the area and perimeter of the Pythagorean triangle. 4. CONCLUSION In this paper, we have presented infinitely many integer solutions for the positive Pell equation 432 22  yx . 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] R.D. Carmichael,The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1950). [2] L.E. Disckson, History of Theory of Numbers, vol II, Chelsea publishing Co., New York (1952). [3] L.J. Mordell, Diophantine Equations, Academic press, London (1969). [4] M.A. Gopalan and R. Anbuselvi, Integral solutions of   1314 22  axaay , Acta Ciencia Indica, XXXIV(1) (2008) 291-295. [5] M.A. Gopalan,et al., Integral points on the hyperbola     0,,2142 222 kakkaayxa  ,IndianJournalof Science, 1(2) (2012) 125-126. [6] M.A. Gopalan, S. Devibala and R. Vidhylakshmi, Integral points on the hyperbola 532 22  yx ,AmearicanJornal of Applied Mathematics and Mathematical Sciences, 1(2012) 1-4. [7] S. Vidhyalakshmi,et al., Observations on the hyperbola   131 22  ayaax , Discovery, 4(10) (2013) 22-24. [8] M.A. gopalan,et al., Integral points on the hyperbola     0,,2142 222 kakkaayxa  , Indian Journal of science, 1(2) (2012) 125-126. [9] M.A. Gopalan and S. Vidhyalakshmi and A. Kavitha, on the integer solutions of binary quadratic equation,   0,,414 222  tkykx t , BOMSR, 2(2014) 42-46. [10] T.R. Usha Rani and K. Ambika, Observation on the Non- Homogeneous Binary Quadratic Diophantine Equation 565 22  yx , Journal of Mathematics and Informatics, Vol-10,Dec (2017), 67-74. [11] M.A. Gopalan and Sharadha Kumar, On the Hyperbola 2332 22  yx , Journal of Mathematics and Informatics, vol-10,Dec (2017),1-9.
  • 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 978 [12] M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha, Observationson the Hyperbola ,13310 22  xy Archimedes J. Math., 3(1) (2013), 31-34. [13] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, On the integral solutions of the binary quadratic equation ,1115 22 t yx  Scholars Journal of Engineering and Technology, 2(2A) (2014), 156-158. [14] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, Observations on the Positive Pell Equation  120 22  xy , International Journal of Pure and Applied Mathematics, 120(6) (2018), 11813-11825.