CHAPTER 2 Quadratic diophantine equations with two unknowns

Transcription

1 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 3 CHAPTER Quadraic diophaie equaio wih wo ukow Thi chaper coi of hree ecio. I ecio (A), o rivial iegral oluio of he biar quadraic diophaie equaio 3X k where k 0, are obaied. No rivial diic ieger oluio of he bi-variae quadraic equaio, are preeed i ecio (B). Secio (C) deal wih he ud of he biar quadraic equaio for i o-ero iegral oluio I each of he ecio, a few iereig relaio amog he oluio, are ehibied. I addiio, he recurrece relaio aified b he oluio alog wih a few eample are illuraed.

2 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 4 SECTION A : The biar quadraic equaio uder coideraio i 3X k (.) To aale he aure of oluio of (.), oe ha o coider differe value for k. We pree below o-ero diic iegral oluio of (.) for he value of k, ad 6. Cae (i): Le k. The correpodig biar quadraic equaio i 3X (.) The malle poiive ieger oluio of (.) i ( X, ) (,) 0 0 To fid he equece of oluio of (.), coider he pellia 3X whoe geeral oluio ( X, ) i give b 3 3 X 3 3, 0,,,3,... 3

3 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 5 Now, applig Brahmagupa lemma [46 ] bewee he oluio ( X0, 0) ad ( X, ), he oher oluio of (.) are repreeed b X Replacig X b, b we oice from he above equaio , 0,,, Treaig, a he rak of diic Triagular umber, oe ma oberve ha T 3 T,,,3,.... 3, 3, The rak, aif he followig recurrece relaio repecivel, Cae (ii): Le k The biar quadraic equaio uder coideraio i 3X

4 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 6 whoe geeral oluio X, i repreeed b X ad 3 3, 0,,,3,... Aumig X ad m we obai 3 3 m ad , 0,,,3,... Coiderig m, a he geeraor of a Phagorea riagle, i leg, ad hpoeue are foud o be m The above value of,, aif he followig recurrece relaio: , 0,,,3..

5 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 7 The followig Table.(a) ehibi a few umerical Phagorea riagle: Table.(a) X ,36 3,408 40,545 46,87 36, ,79 65,08 4,54,60 7,865,5 9,08,3 63,50,08 09,55,575 6,500,47 880,96,760,55,870,59,76,93,5 Z We pree below a few iereig relaio amog he ide i each of he above Phagorea riagle:. The Phagorea riagle are primiive.. Deoig he perimeer of he Phagorea riagle b P we obai P 0mod. 3. P ca be epreed a he differece of wo quare. 4. P3 0(mod0) P3 5. 0(mod0) 6. Deoig he Area of he Phagorea riagle b we obai A A 0(mod0). 7. mod.

6 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS mod (mod8) (mod5). P3 0(mod0) P3. 0(mod0) 3. The riple,, obaied from he liear raformaio X, X, aif he erar cubic equaio From each of he value of,,3,... ecod order Ramauja umber., oe ca geerae = = Here we oberve ha 30,00,96,338,90,70 are ecod order Ramauja umber.

7 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 9 Cae (iii): Le k 6. I hi cae, coidered biar quadraic equaio i 3X 6 (.3) The malle poiive ieger oluio of (.3) i ( X, ) (,3) 0 0 To fid he oher oluio of (.3), coider he pellia 3X whoe geeral oluio ( X, ) i give b 3 3 X 3 3, 0,,,3,... 3 Now, applig Brahmagupa lemma [46] bewee he oluio ( X0, 0) ad ( X, ), he oher oluio of (.3) are repreeed b X

8 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 30 Replacig X b, b we oberve from he above equaio , 0,,, Treaig, a he rak of Ceered Triagular umber ad Triagular umber, oe ma oberve ha CT3, T3,,,,3,... The rak, aif he followig recurrece relaio repecivel,

9 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 3 SECTION B: Coider he biar quadraic Diophaie equaio (.4) The hree differe paer of oluio of (.4) are preeed below: Paer : The ubiuio of he liear raformaio u v, u v i (.4), lead o which i aified b u u(4v) ( v ) 0 u ad v {[(9 4 5) (3 5) (9 4 5) (3 5)] } 5 [(9 4 5) ( 5 3) (9 4 5) ( 5 3)] 5 [(9 4 5) (3 5) (9 4 5) (3 5) 4], =0,,4,... 0

10 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 3 Kowig he value of u ad v, he oluio of (.4) are obaied a [(9 4 5) ( 7 5 5) (9 4 5) ( 7 5 5) 6 5] 0 5 ( or) [(9 4 5) (3 5 5) (9 4 5) (3 5 5) 5], 0 5 =0,,4,... [(9 4 5) (3 5)(3 5 5) (9 4 5) (3 5)(3 5 5) 6 5] ( or) (9 4 5) (3 5)(3 5 5) (9 4 5) (3 5)(3 5 5) 5] 0 5 A few umerical eample are preeed below i Table.(a): Table.(a) 6,-44-0, , , , , , , Paer : Treaig equaio (.4) a a quadraic i ad olvig for, we ge Aume [ ( ) 5 9] (.5) 5 9

11 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 33 The above equaio i wrie a 5 44 (.6) where 5 (.7) The malle poiive ieger oluio of (.6) i 4, To obai he oher oluio of (.6), coider he Pellia 5 whoe geeral oluio (, ) i give b 5 (94 5), 0,,,... Sice irraioal roo occur i pair, we have 5 (9 4 5), 0,,,... From he above wo equaio, we have [(9 4 5) (9 4 5) 5 ad [(9 4 5) (9 4 5), 0,,,...

12 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 34 Applig Brahmagupa Lemma bewee he oluio ( 0, 0) ad (, ) he oluio of (.6) i give b, 0 0 [(9 4 5) ( 5 3) (9 4 5) ( 5 3)] 5 5 (94 5) (3 5) (94 5) (3 5) 0 0 I view of (.7), we have Thu 5 [(9 4 5) (3 5) (9 4 5) (3 5) ] (.8) 5 5 Subiuig (.8) i (.5), he correpodig value of are [(9 4 5) (3 5) (9 4 5) (3 5) 6] 0 [(9 4 5) ( 5 3) (9 4 5) ( 5 3)], 0,, 4,... 5 (.9) Equaio (.9) ad (.8) are oluio of (.4). A few umerical oluio are ehibied i he followig Table.(b): Table.(b): 6, , , ,

13 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 35 I i oberved ha (5 ) 5( ) 44 P aer 3: Treaig equaio (.4) a a quadraic i ad olvig for, we ge Aume [( ) 5 6 7] (.0) 5 67 The above equaio i wrie a where 5 44 (.) 53 (.) The malle poiive ieger oluio of (.) i 4, To obai he oher oluio of (.), coider he Pellia 5 whoe geeral oluio (, ) i give b 5 (94 5), 0,,,... Sice irraioal roo occur i pair, we have 5 (94 5), 0,,,....

14 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 36 From he above wo equaio, we have [(9 4 5) (9 4 5) 5 ad [(9 4 5) (9 4 5), 0,,,... Applig Brahmagupa lemma bewee he oluio (, ) 0 0 ad (, ), he oluio of (.) are give b 0 0 [(9 4 5) ( 5 4) (9 4 5) ( 5 4)] (94 5) (4 5) (94 5) (4 5) I view of (.), we have Thu [(9 4 5) (4 5) (9 4 5) (4 5) 3] (.3) 5 5 Subiuig (.3) i (.0), he correpodig value of are [(9 4 5) (4 5) (9 4 5) (4 5) ] 0 [(9 4 5) ( 5 4) (9 4 5) ( 5 4)],,3,5,... 5 (.4) Equaio (.3) ad (.4) are oluio of (.4).

15 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 37 A few umerical oluio of (.4) are give i Table.(c): I i ee ha Table.(c): , , , , O bervaio: (5 3) 5( ) 44. I i oed ha he Diophaie equaio (.4) repree a hperbola ad hu, he above oluio paer repree he iegral poi o he hperbolic curve.. I paer ad, he -value are he ame wherea he -value are differe. 3. If (, ) i a iegral oluio of he give equaio (.4) he 0 0, 3 ) ( i alo a oluio of (.4). 4. The oluio ad of equaio (.4) aif he followig recurrece relaio repecivel: ad

16 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 38 S ECTION C: The Diophaie equaio of he ile 63 0 (.5) repree a hperbola. B hifig he origi o he cere (-,-3), he equaio (.5) reduce o X (.6) w here X, =+3 (.7) Equaio (.6) i he well kow Pellia equaio whoe geeral oluio ( X, ) i give b X (0 3 ), =0,,,... (.8) i whic h ( 0, 0) (3,0) i he mall poiive ieger oluio of (.6). Sice irraioal roo occur i pair, we have X (0 3 ) (.9) From (.8) ad (.9), he value of X, are obaied a follow: X [(0 3 ) 3 ) (0 ] [(0 3 ) (0 3 ) ]

17 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS 39 I view of (.7), he o-ero iegral oluio of (.7) are give b [(0 3 ) (0 3 ) ] [(0 3 ) (0 3 ) 6], =0,,,3,... (.0) A few umerical eample are preeed i he below able.3(a): Table.3(a): The recurrece relaio aified b ehibied below:, are correpodigl A few iereig relaio oberved amog he oluio (.0) are preeed below:. 0 (mod 4) (mod ) 0 (mod )

18 CHAPTER - QUADRATIC DIOPHANTINE EQUATIONS WITH TWO UNKNOWNS (mod ) 5. 0 (mod ) ( 6)( 5) 0 ( mod6) i a cubical ieger 3. The followig epreio (i) 6[ 8] (ii) 6[ 4] repree a a umber.