IRJET- On the Pellian Like Equation 5x2-7y2=-8
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This document analyzes the binary quadratic equation 875 - 22 = -yx for its distinct integer solutions. Some key findings include: 1) The equation has an infinite number of integer solutions that can be generated recursively. 2) Interesting relationships are found among the solutions. 3) Knowing a solution to the original equation, solutions can be found for other hyperbolic and parabolic equations.
![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 979
On the Pellian Like Equation 875 22
yx
S. Vidhayalakshmi1, A. Sathya2, S. Nivetha3
1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.
2Assisant 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.
----------------------------------------------------------------------------***-------------------------------------------------------------------
Abstract – The binary quadratic equation represented by the
pellian like equation 875 22
yx is analyzedforitsdistinct
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 Diophantine equation 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 with the problemofobtaining
non-zero distinct integer solutions to the binary quadratic
equationgiven by 875 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
875 22
yx (1)
Consider the linear transformations
TXyTXx 57 (2)
From (1) and (2), we have
435 22
TX (3)
whose smallest positive integer solution is
212 00 TX
To obtain the other solutions of (3), consider the pellian
equation is
135 22
TX (4)
whose smallest positive integer solution is
)1,6()
~
,
~
( 0 oTX
The general solution of (4) is given by
nn gT
352
1~
, nn fX
2
1~
where
11
356356
nn
nf
11
356356
nn
ng
Applying Brahmagupta lemma between ),( 00 TX and
)
~
,
~
( nn TX the other integer solutions of (3) are given by
nnn
nnn
gfT
gfX
35
6
356
1
1
(5)
From (2), (4) and (5) the values of x and y satisfying (1) are
given by
nnn gfx
35
77
131 ](https://image.slidesharecdn.com/irjet-v6i3180-190814062650/85/IRJET-On-the-Pellian-Like-Equation-5x2-7y2-8-1-320.jpg)
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IRJET- On the Pellian Like Equation 5x2-7y2=-8
- 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 979 On the Pellian Like Equation 875 22 yx S. Vidhayalakshmi1, A. Sathya2, S. Nivetha3 1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India. 2Assisant 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. ----------------------------------------------------------------------------***------------------------------------------------------------------- Abstract – The binary quadratic equation represented by the pellian like equation 875 22 yx is analyzedforitsdistinct 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 Diophantine equation 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 with the problemofobtaining non-zero distinct integer solutions to the binary quadratic equationgiven by 875 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 875 22 yx (1) Consider the linear transformations TXyTXx 57 (2) From (1) and (2), we have 435 22 TX (3) whose smallest positive integer solution is 212 00 TX To obtain the other solutions of (3), consider the pellian equation is 135 22 TX (4) whose smallest positive integer solution is )1,6() ~ , ~ ( 0 oTX The general solution of (4) is given by nn gT 352 1~ , nn fX 2 1~ where 11 356356 nn nf 11 356356 nn ng Applying Brahmagupta lemma between ),( 00 TX and ) ~ , ~ ( nn TX the other integer solutions of (3) are given by nnn nnn gfT gfX 35 6 356 1 1 (5) From (2), (4) and (5) the values of x and y satisfying (1) are given by nnn gfx 35 77 131
- 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 980 nnn gfy 35 65 111 The recurrence relation satisfied by the solution x and y are given by 012 123 nnn xxx 012 123 nnn yyy Some numerical examples of nx and ny satisfying (1) are given in the Table :1 below Table: 1 Numerical Examples n nx ny 0 26 22 1 310 262 2 3694 3122 3 44018 37202 4 524522 443302 5 6250246 5282422 6 74478430 62945762 7 887490914 750066722 From the above table, we observe some interesting relations among the solutions which are presented below: nx and ny values are always even. 1. Relation among the solutions are given below: 067 112 nnn xyx 07184 113 nnn xyx 056 112 nnn xyy 06071 113 nnn xyy 012 123 nnn xxx 076 221 nnn yxx 014 213 nnn yxx 06717 123 nnn xxy 08471 331 nnn yxx 076 223 nnn yxx 06571 312 nnn yxy 06717 321 nnn xxy 065 221 nnn yxy 010 321 nnn yxy 065 332 nnn yxy 05671 312 nnn xyy 07160 331 nnn yxy 012 321 nnn yyy 067 233 nnn xxy 056 223 nnn xyy 2. Each of the following expressions represents a nasty number: 465773 2222 nn xy 2891777 7 3 2232 nn xx 3361092777 28 1 2242 nn xx 2477577 2 1 2232 nn xy 284923577 71 3 2242 nn xy 2465917 2 1 3222 nn xy 2846510927 71 3 4222 nn xy 4131553 3222 nn yy 48131847 4 1 4222 nn yy 415611313 3242 nn xx 47759173 3232 nn xy 249235917 2 1 3242 nn xy 2477510927 2 1 4232 nn xy
- 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 981 49235109273 4242 nn xy 415518473 4232 nn yy 3. Each of the following expressions representsacubical integer: 113333 1952316577 2 1 nnnn xyxy 1 23343 2751 23191777 14 1 n nnn x xxx 1 33353 32781 2311092777 168 1 n nnn x xxx 1 23343 2325 23177577 12 1 n nnn x yxy 1 33353 27705 231923577 142 1 n nnn x yxy 2 14333 195 275165917 12 1 n nnn x yxy 3 15333 195 327816510927 142 1 n nnn x yxy 214333 3946513155 2 1 nnnn yyyy 315333 395541131847 24 1 nnnn yyyy 234353 46833931561131 2 1 nnnn xxxx 2 24343 2325 2751775917 2 1 n nnn x yxy 2 34353 27705 27519235917 12 1 n nnn x yxy 3 25343 2325 3278177510927 12 1 n nnn x yxy 3 35353 27705 32781923510927 2 1 n nnn x yxy 3 25343 465 55411551847 2 1 n nnn y yyy 4. Each of the following expressions represents a biquadratic integer: 12260 3086577 2 1 22 224444 n nnn x yxy 843668 30891777 14 1 22 324454 n nnn x xxy 100843708 3081092777 168 1 22 424464 n nnn x xxy 723100 30877577 12 1 22 324454 n nnn x yxy 85236940 308923577 142 1 22 424464 n nnn x yxy 72260 366865917 12 1 32 225444 n nnn x yxy 85226065 4370810927 142 1 4264 2244 nn nn xx yy 1252 62013155 2 1 32 225444 n nnn y yxy 14452 7388131847 24 1 42 226444 n nnn y yyy 8443708 366810927917 14 1 32 425464 n nnn x xxx
- 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 982 123100 3668775917 2 1 32 325454 n nnn x yxy 7236940 36689235917 12 1 32 425464 n nnn x yxy 723100 4370877510927 12 1 42 326454 n nnn x yxy 1236940 43708923510927 2 1 42 426464 n nnn x yxy 12620 73881551847 2 1 42 326454 n nnn y yyy 5. Each of the following expressions represents aquintic integer: 11 33335555 650770 3253856577 2 1 nn nnnn xy xyxy 1233 435565 91707704585 38591777 14 1 nnn nnn xxx xxx 1333 535575 10927077054635 3851092777 168 1 nnn nnn xxx yxx 1233 435565 77507703875 38577577 12 1 nnn nnn xyx yxy 1333 535575 9235077046175 385923577 142 1 nnn nnn xyx yxy 2143 336555 6509170325 458565917 12 1 nnn nnn xyx yxy 3153 337555 650109270325 546356510927 142 1 nnn nnn xyx yxy 2143 336555 130155065 77513155 2 1 nnn nnn yyy yyy 3153 337555 1301847065 9235131847 24 1 nnn nnn yyy yyy 2343 536575 109270917054635 458510927917 14 1 nnn nnn xxx xxx 2243 436565 775091703875 4585775917 2 1 nnn nnn xyx yxy 2343 536575 923500917046175 45859235917 12 1 nnn nnn xyx yxy 3253 437565 77501092703875 5463577510927 12 1 nnn nnn xyx yxy 3353 537575 9235010927046175 54635923510927 2 1 nnn nnn xyx yxy 3253 437565 155018470775 92351551847 2 1 nnn nnn yyy yyy 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 S.NO Hyperbola (X,Y) 1 1635 22 XY 11 11 6577 ,1311 nn nn xy yx 2 78435 22 XY 12 21 91777 ,13155 nn nn xx xx 3 11289635 22 XY 13 31 1092777 ,131847 nn nn xx xx 4 57635 22 XY 12 11 77577 ,13131 nn nn xy yx 5 8065635 22 XY 13 31 923577 ,131561 nn nn xy yx
- 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 983 6 57635 22 XY 21 12 65917 ,15511 nn nn xy yx 7 8065635 22 XY 31 13 6510927 ,184711 nn nn xy yx 8 40035 22 XY 21 12 65775 ,13111 nn nn yy yy 9 5760035 22 XY 31 13 659235 ,156111 nn nn yy yy 10 78435 22 XY 23 32 10927917 ,1551847 nn nn xx xx 11 1635 22 XY 22 22 775917 ,155131 nn nn xy yx 12 57635 22 XY 23 32 9235917 ,1551561 nn nn xy yx 13 57635 22 XY 32 23 77510927 ,1847131 nn nn xy yx 14 1635 22 XY 33 33 923510927 ,18471561 nn nn xy yx 15 40035 22 XY 32 23 7759235 ,1561131 nn nn yy yy II. Employing linear combinationsamongthesolutionsof (1), one may generate integer solutions for other choices of parabolas which are presented in Table: 3 below: Table: 3 Parabolas S.NO Parabola (X,Y) 1 16352 2 XY 46577 ,1311 2222 11 nn nn xy yx 2 11252 2 XY 2891777 ,13155 2232 21 nn nn xx xx 3 16128524 2 XY 3361092777 ,131847 2242 31 nn nn xx xx 4 5763512 2 XY 2477577 ,13131 2232 11 nn nn xy yx 5 8065635142 2 XY 284923577 ,131561 2242 31 nn nn xy yx 6 5763512 2 XY 2465917 ,15511 3222 12 nn nn xy yx 7 8065635142 2 XY 2846510927 ,184711 4222 13 nn nn xy yx 8 8072 2 XY 2065775 ,13111 3222 12 nn nn yy yy 9 115207120 2 XY 48131847 ,156111 4222 13 nn nn yy yy 10 11252 2 XY 2810927917 ,1551847 3242 32 nn nn xx xx 11 16352 2 XY 4775917 ,155131 3232 22 nn nn xy yx 12 5763512 2 XY 249235917 ,1551561 3242 32 nn nn xy yx 13 5763512 2 XY 2477510927 ,1847131 4232 23 nn nn xy yx 14 16352 2 XY 4923510927 ,18471561 4242 33 nn nn xy yx 15 8072 2 XY 207759235 ,1561131 4232 23 nn nn yy yy III. Let qp, be two non-zero distinct integers such that 0 qp . Treat qp, as the generators of the Pythagorean triangle ),,( T
- 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 984 where pq2 , 22 qp , 22 qp , 0 qp Taking 11 nn yxp , 1 nyq , it is observed that ),,( T is satisfied by the following relations: 161073 p A 4 112 nn yx p A Where PA, represent the area and perimeter of ),,( T . 3. CONCLUSION In this paper, we have presented infinitely many integer solutions for the Pellian like equation 875 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 ,IndianJournal of 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 ,IndianJournalof 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. [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