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Pythagorean Equation and Special M-Gonal Numbers IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 1 (Sep. - Oct. 2013), PP 31-34 www.iosrjournals.org Pythagorean Equation and Special M-Gonal Numbers M. A. Gopalan1, V. Geetha2 1(Department of Mathematics, Shrimathi Indira Gandhi College,Trichirappalli-620 002.) 2(Department of Mathematics, Cauvery College For Women,Trichirappalli-620 018.) Abstract: Employing the solutions of the Pythagorean equation, we obtain the relations between the pairs of special polygonal numbers such that the difference in each pair is a perfect square. Key Words: Pythagorean equation,Polygonal numbers MSC Classification Number: 11D09. NOTATIONS PenP = Pentagonal number of rank P HexQ = Hexagonal number of rank Q HepH = Heptagonal number of rank H DodecD = Dodecagonal number of rank D I. Introduction In [1,2,4-8,10],employing the integral solutions of special binary quadratic Diophantine equation, special patterns of Pythagorean triangles are generated. In [3],the relations among the pairs of special m-gonal numbers generated through the solutions of the binary quadratic equation yx2221 are determined. In [9], the relations among special figurate numbers through the equation yx2210 1 are obtained. In[11],employing the solutions of the Pythagorean equation, and obtain the relations between Triangular number and Pentagonal number, Octagonal number, Hexagonal number, Heptagonal number, Decagonal number, Dodecagonal number , Pentagonal number and Hexagonal number, Octagonal number such that the difference in each pair is a perfect square. In this communication, employing the solutions of the Pythagorean equation, we obtain the relations between the pairs of special polygonal numbers which are not mentioned in [11] such that the difference in each pair is a perfect square. II. Method Of Analysis Consider the Pythagorean equation x2 y 2 z 2 (1) whose solutions are 2 2 2 2 x2 rs , y r s , z r s (2) where rs, are non – zero distinct integers and rs . Case.2.1: The choice 2 2 2 2 10H 3 r s ,18 P 3 r s (3) in (1) leads to the relation 2 40HepHP 216 Pen (4) From (3), the values of ranks of the Heptagoanl numbers and Pentagonal numbers are respectively given by r2 s 2 33 r 2 s 2 PH; 18 10 It is seen that PH, are integers for the following choices of r and s namely, (i) sr1; 14 (ii) sr4; 11 (iii) s15 n 14; r 15 n 11 www.iosrjournals.org 31 | Page Pythagorean Equation And Special M-Gonal Numbers (iv) s15 n 4; r 15 n 11 (v) s15 n 4; r 15 n 29 For each of the values of and the corresponding Pentagonal and Heptagonal numbers satisfying (4) are presented in the Table (1)below. Table (1) S . α N P PenP HepH o 1 6 14 51 469 88 2 11 20 176 970 28 1 2 1 22 2 3 54n 45nn 75 32 (75nn2 125 52) (45n 75 n 32)(225 n 375 n 157)(450nn 750 308) 2 2 1 1 222 4 25n 6 2 (1875nn2 875 102) (45n 21 n 14)(225 n 105 n 67)(450nn 210 88) 45nn 21 14 2 2 H 1 2 1 55n 46 2 (9075nn 15125D 6302) (45n22 75 n 86)(225 n 375 n 427) 2 5 45nn 75 86 2 (450nn 750 232) 2 Case 2.2: The Choice 5D 2 r2 s 2 ;12 P 2 r 2 s 2 (5) in (1) leads to the relation 2 96PenPD 5 Dodec From (5), the values of ranks of the Pentagonal and Dodecagonal numbers are respectively given by, r2 s 2 22 r 2 s 2 PD, 12 5 It is seen that P and are integers for the following choices of and namely, (i) s1; r 30 n 27 (ii) s9; r 30 n 7 (iii) s9; r 30 n 17 (iv) s1; r 30 n 3 (v) s11; r 30 n 3 (vi) s19; r 30 n 3 (vii) s30 n 19; r 60 n 27 (viii) s30 n 1; r 60 n 3 For each of the values of and the corresponding values of and are presented in the Table (2) below. Table (2) S.No P D 2 2 1 75nn 135 61 180nn 324 146 2 2 2 75nn 35 11 180nn 84 6 2 2 3 75nn 85 31 r s 180nn 204 42 2 2 4 75nn 15 1 180nn 36 2 2 2 5 75nn 15 11 180nn 36 22 6 2 2 75nn 15 31 180nn 36 70 www.iosrjournals.org 32 | Page Pythagorean Equation And Special M-Gonal Numbers 7 2 2 375nn 365 91 540nn 420 74 8 2 2 375nn 35 1 540nn 60 2 In Table(3) below represent the corresponding Pentagonal and Dodecagonal numbers are exhibited. Table(3) S. α No DodecD 1 22 22 1 (75n 135 n 61)(225 n 405Pen nP 182) (180n 324 n 146)(900 n 1620 n 726) (60n 54) 2 1 2 (75n22 35 n 11)(225 n 105 n 32) (180n22 84 n 6)(900 n 420 n 34) (540n 126) 2 22 1 22 (180n 204 n 42)(900 n 1020 n 206) 3 (75n 85 n 31)(225 n 255 n 92) (540n 306) 2 1 4 (75n22 15 n 1)(225 n 45 n 2) (180n22 36 n 2)(900 n 180 n 6) (60n 6) 2 1 5 (75n22 15 n 11)(225 n 45 n 32) (180n22 36 n 22)(900 n 180 n 114) (660n 66) 2 H 1 6 (75n22 15 n 31)(225 n 45 n 92) (180n22 36 n 70)(900 n 180 n 354) (1140n 114) 2 22 2 1 22 (3600nn 3900 1026) 7 (375n 365 n 91)(1125 n 1095 n 272) (540n 420 n 74)(2700 n 2100 n 366) 2 2 1 22 22(3600nn 300 6) 8 (375n 35 n 1)(1125 n 105 n 2) (540n 60 n 2)(2700 n 300 n 6) 2 Case2.3: The Choice 12Q 3 r2 s 2 ;10 H 3 r 2 s 2 (6) in (1) leads to the relation 2 72HexQH 40 Hep . From (6), the values of ranks of the Hexagonal and Heptagonal numbers are respectively given by r2 s 2 33 r 2 s 2 QH; 12 10 which are integers for the following three choices of and namely, (i) s15 n 3; r 15 n 6 (ii) s15 m 12; r 30 n 15 m 21 (iii) s15 m 12; r 30 n 15 m 39 For each of the values of and the values of Q and are presented in the Table(4) below Table(4) S.No 1 2 1 (450nn 90 48) 27n 3 12 1 22 2 (900n 900 nm 450 m 1260 n 990 m 588) 90n2 90 nm 126 n 27 m 30 12 1 22 3 (900n 900 nm 450 m 2340 n 1170 m 1668) 90n2 90 nm 234 n 81 m 138 12 r s www.iosrjournals.org 33 | Page Pythagorean Equation And Special M-Gonal Numbers In Table(5) below represent the corresponding Hexagonal and Heptagonal numbers are exhibited. Table(5) III. Conclusion One may search for relations among other m-gonal numbers such that the difference in each pair is a perfect square. References 22 [1] M.A.Gopalan and G.Janaki,Observations on yx31Acta Ciencia Indica XXXIVM (2) (2008) 103-106. 22 [2] M.A.Gopalan and B.Sivakami,Observations on the integral solutions of yx71, Antarctica J. Math. 7(3),(2010) ,291- 296 22 [3] M.A.Gopalan and G.Srividhya,Relations among m-gonal numbers through equation yx21,Antarctica J. Math. 7(3),(2010) ,363-369 [4] M.A.Gopalan and R.Vijayalakshmi,Special Pythagorean triangles generated through the integral solutions of the equation 2 2 2 y( k 1) x 1,Antarctica J. Math. 7(5),(2010), 503-507 22 [5] M.A.Gopalan and R.Vijayalakshmi, Observations on the integral solutions of yx51 , Impact J.Sci.Tech. 4(4),(2010), 125-129 2 2 2 [6] M.A.Gopalan and R.S.Yamuna, Remarkable observations on the binary quadratic equation y( k 2) x 1 , Impact J.Sci.Tech. 4(4),(2010), 61-65 22 [7] M.A.Gopalan and G.Sangeetha,A Remarkable observation on the binary quadratic equation yx10 1 , Impact J.Sci.Tech. 4 (2010), 103-106 22 [8] M.A.Gopalan and R.Palanikumar, Observations on yx12 1 , Antarctica J. Math. 8(2),(2011),149-152 22 [9] Manju Somanath,G.Sangeetha and M.A.Gopalan, Relations among special figurate numbers through equation yx10 1 , Impact J.Sci.Tech. 5(1),(2011), 57-60 22 [10] M.A.Gopalan and K.Geetha, Observations on the Hyperbola yx18 1,RETELL, 13(1), Nov.2012,81-83. [11]. M.A.Gopalan, V.Sangeetha and Manju Somanath, Pythagorean equation and Special m-gonal numbers, sent to Antartica journal www.iosrjournals.org 34 | Page .
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