Pythagorean Triangle with Area/ Perimeter as a special polygonal number

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: p-ISSN: X Volume 7 Issue 3 (Jul. - Aug. 013) PP 5-6 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber M.A.Gopala 1 Maju Somaath K.Geetha 3 1. Departmet of Mathematics Shrimati Idira Gadhi College Trichy.. Departmet of Mathematics Natioal College Trichy. 3. Departmet of Mathematics Cauvery College for Wome Trichy. Abstract: Patters of Pythagorea triagles i each of which the ratio Area/ Perimeter is represeted by some polygoal umber. A few iterestig relatios amog the sides are also give. Keyword: Polygoal umber Pyramidal umber Cetered polygoal umber Cetered pyramidal umber Special umber I. Itroductio The method of obtaiig three o-zero itegers x y ad z uder certai coditios satisfyig the relatio x y z has bee a matter of iterest to various Mathematicias [ ]. I [7-15] special Pythagorea problems are studied. I this commuicatio we preset yet aother iterestig Pythagorea problem. That is we search for patters of Pythagorea triagles where i each of which the ratio Area/ Perimeter is represeted by a special polygoal umber. Also a few relatios amog the sides are preseted. Notatio m p = Pyramidal umber of rak with sides m t m = Polygoal umber of rak with sides m jal = Jacobsthal Lucas umber ja = Jacobsthal umber ct m = Cetered Polygoal umber of rak with sides m m cp = Cetered Pyramidal umber of rak with sides m g = Gomoic umber of rak with sides m p = Proic umber carl = Carol umber ky =Kyea umber II. Method of Aalysis The most cited solutio of the Pythagorea equatio x y z ( is represeted by x uv y u v z u v u v 0 () Patter 1: Deotig the Area ad Perimeter of the triagle by A ad P respectively the assumptio A t11 leads to the equatio q p q 9 7 This equatio is equivalet to the followig two systems I ad II respectively: 5 Page

2 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber 9-7 N N 9-7 I what follows we obtai the values of the geerators p q ad hece the correspodig sides of the Pythagorea triagle O evaluatio the values of the geerators satisfyig system I are p 10 7 q Employig () the sides of the correspodig Pythagorea triagle are give by 0 14 y x z X Y Z A P z y x 8 6 is a Nasty umber[] t 10x y z 4 mod140 ) 10x 1 y 1 z t3 1 3) Case : O evaluatio the values of the geerators satisfyig system II are p 10 7 q 9 7 Usig () the correspodig Pythagorea triagle is y x z X Y z A P z 9y s t 97mod ) 10y z x 5t65 p 5 98 mod 8 3) 10x 9 y z 98mod140 Patter : The assumptio leads to the equatio A t1 5 4 q p q This equatio is equivalet to the followig two systems I ad II respectively 53 Page

3 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of the geerators satisfyig system I are p7 4 q Usig () the correspodig Pythagorea triagle is x( ) 8 16 y( ) z( ) X Y z A P xz y 8t 0mod88 58 ) x y z 4t1 0 3) 8 6 y is a asty umber z Case : O evaluatio the values of the geerators satisfyig the system II are p7 4 q6 4 Usig () the correspodig Pythagorea triagle is x( ) y( ) 13 8 z( ) x Y z A P z x y ct t mod ) x 6y 3g 8 is a Nasty umber x 1 6y 1 1t mod 6 3) Patter 3: Uder our assumptio leads to the equatio A t q p q This equatio is equivalet to the followig two systems I ad II 54 Page

4 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber 11-7 N 11-7 O evaluatio the values of the geerators satisfyig system I are p1 7 q I view of () the correspodig Pythagorea triagle is x( ) 4 14 y( ) z( ) x Y z A P y is a Nasty umber z ) z 6x t3 49mod85 3) z y x 6p 4 t3 0mod 4 Case : O evaluatio the values of the geerators satisfyig system II are p11 7 q10 7 Employig () the correspodig Pythagorea triagle is x y 1 14 z x Y z A P x 10y t 98mod145 ) z 10y 6p 13 5t8 t3 0 mod ) z xx 13 y t cp 506 p Patter 4: Uder the assumptio A t14 leads to the equatio q p q 6 5 This equatio is equivalet to the followig two systems Page

5 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p8 5 q Employig () the correspodig Pythagorea triagle is x 3 0 y z x Y z A P z x t9 5mod37 z y 8cp 3) x 6cp 30 p 8 t44 0mod5 Case : O evaluatio the values of geerators satisfyig II are p8 5 q6 5 Employig () the correspodig Pythagorea triagle is x( ) y( ) 8 0 z( ) ) 6 x Y z A P x3y 0g 5 is a Nasty umber ) z ct3 3mod116 3) x 3y 3 6ct8 g Patter 5: The assumptio A t15 leads to the equatio q p q The above equatio is equivalet to the followig two systems I ad II respectively 56 Page

6 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber N As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p14 11 q Employig () the correspodig Pythagorea triagle is x( ) 8 y( ) z( ) x Y z A P z 7x 4cp 196cp 11 is a perfect square ) z y is biquadratic iteger 3) x 13 6cp 3t4 3 is cubic iteger Case : O evaluatio the values of geerators satisfyig system II are p14 11 q13 11 Employig () the correspodig Pythagorea triagle is x( ) y( ) 7 z( ) x Y z A P z x x 13 y t 561 cp 506 p z x y t56 4cp 14y z t15 4 mod 75 Patter 6: The assumptio A t16 leads to the equatio q p q 7 6 ) 6 3) The above equatio is equivalet to the followig two systems I ad II respectively 57 Page

7 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p9 6 q Employig () the correspodig Pythagorea triagle is x( ) 36 4 y( ) z( ) x y z A P x 3y t58 108mod186 ) x t58 8 0mod3 3) z y ct16 1mod8 Case : O evaluatio the values of geerators satisfyig equatio II are p9 6 q7 6 Employig () the correspodig Pythagorea triagle is x( ) y( ) 3 4 z( ) x y z A P y x z 48g 4 is a Nasty umber ) y( x ct4 65(mod 94) 3) z( x ct8 13g 189 Patter 7: The assumptio A t17 leads to the equatio q p q The above equatio is equivalet to the followig two systems I ad II respectively N Page

8 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p16 13 q Employig () the correspodig Pythagorea triagle is x( ) 3 6 y( ) z( ) x y z A P x( ) 8 jal 6mer 34 0 ) y z car l ky Case : O evaluatio the values of geerators satisfyig system II are give by p16 13 q15 13 Employig () the correspodig Pythagorea triagle is x( ) y( ) 31 6 z( ) x y z A P y31p 0mod57 ) y z x t 6 0 mod3 3) x 15y z x 4 ct. ct 3t 7t 4 mod 61 Patter 8: Uder our assumptio A t18 leads to the equatio q p q The above equatio is equivalet to the followig two systems I ad II respectively As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are 59 Page

9 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber p10 7 q Employig () the correspodig Pythagorea triagle is x( ) 40 8 y( ) z( ) x y z A P x t y x Rh t ) z y x z 18t6 141g Case : O evaluatio the values of geerators satisfyig system I are ) p10 7 q Employig () the correspodig Pythagorea triagle is x( ) y( ) 36 8 x y z A P z x 1 ct3 t11 1 5x y z 76g 4t 169 ) 50 x 4y 1 t t 80g ) 17 3 Patter 9: Uder our assumptio A t19 leads to the equatio q p q z( ) The above equatio is equivalet to the followig two systems I ad II respectively N As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p0 17 q Employig () the correspodig Pythagorea triagle is 60 Page

10 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber x( ) y( ) z( ) x y z A P z y jal 3ja ) xt 0mod 4 4 z y 1 t ct g 0 Case : O evaluatio the values of geerators satisfyig system I are 3) p0 17 q19 17 Employig () the correspodig Pythagorea triagle is x( ) y( ) z( ) x y z A P y 8car l 11ky 39 ) 0y z t1 74g z x y 0 mod34 3) Patter 10: Uder our assumptio A t0 leads to the equatio q p q 5 4 The above equatio is equivalet to the followig two systems I ad II respectively As i the previous case we obtai the values of the geerators u v ad hece the correspodig sides of the Pythagorea triagle. O evaluatio the values of geerators satisfyig system I are p7 4 q Employig () the correspodig Pythagorea triagle is x( ) 8 16 y( ) z( ) Page

11 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber x y z A P z y x 45 cp 6 t p ) 3y x z t 3mod ) x z y 1 49g 8 is a asty umber t t Case : O evaluatio the values of geerators satisfyig system II are p7 4 q5 4 Employig () the correspodig Pythagorea triagle is x( ) y( ) 4 16 z( ) x y z A P x z 6y 48g 16 ) z x t3 1 3) z 3y ct 31mod50 4 III. Coclusio Oe may search for other patters of Pythagorea triagles uder cosideratio. Refereces [1]. Albert H.Beiler Recreatios i the Theory of Numbers (Dover Publicatios New York 1963). []. Bhatia B.L. ad Supriya Mohaty Nasty umbers ad their characterizatios (Mathematical Educatio P July Sep 1985). [3]. Dickso L.E. History of the Theory of umber (Chelesa publishig Compay New York Vol II 195). [4]. Malik S.B. Basic Number theory (Vikas Publishig house pvt Ltd. New Delhi 1998). [5]. Mordell L.J. Diophatie equatios (Academic press New York 1969). [6]. Iva Nive Zuckerma Herbert.S ad Motgomery Hugh.L A itroductio to the Theory of Numbers (Joh Wiley ad Sos Ic New York 004). [7]. Gopala M.A. ad Devibala.S. Pythagorea Triagle: A Tressure house proceedig of the KMA atioal semiar o Algebra Number theory ad applicatios to codig ad Cryptaalysis Little Flower College Guruvayur 004 Pp [8]. Gopala M.A. ad Abuselvi.R. A Special Pythagorea Triagle Acta Ciecia Idica VolXXXIM No Pp.53. [9]. Gopala M.A. ad Devibala.S. O a Pythagorea Triagle Problem Acta Ciecia idica Vol.XXXIIM No Pp [10]. Gopala M.A. ad Gaam.A A Special Pythagorea Triagle Acta Ciecia Idica VolXXXIII M No Pp [11]. Gopala M.A. ad Leelavathi.S. Pythagorea triagle with Area/Perimeter as a cubic iteger Bulleti of Pure ad Applied Scieces Vol.6E No. 007 Pp [1]. Gopala M.A. ad Sriram.S. Pythagorea triagle with Area/ Perimeter as differece of two squares Impact joural of Sciece ad Techology Vol. No Pp [13]. Gopala M.A. ad Jaaki G. Pythagorea traiagle with Area/Perimeter as a special polygoal umber Bulleti of Pure ad Applied Scieces Vol.7E No. 008 Pp [14]. Gopala M.A. ad Leelavathi S. Pythagorea triagle with Area/Perimeter as a Square iteger Iteratioal joural of Mathematics Computer Scieces ad Iformatio Techology Vol.1 No. July- Dec 008 Pp [15]. Gopala M.A. ad Vijayasakar.A. Observatios o a Pythagorea problem Acta Ciecia Idica Vol-XXXVIM No Pp Page