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

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 yx2221 are determined. In [9], the relations among special figurate numbers through the equation yx2210 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 x2 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) sr1; 14 (ii) sr4; 11 (iii) s15 n  14; r  15 n  11

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(iv) s15 n  4; r  15 n  11 (v) s15 n  4; r  15 n  29 For each of the values of r and s the corresponding Pentagonal and Heptagonal numbers satisfying (4) are presented in the Table (1)below.

Table (1) S . α N P H 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

1 2 1 55n  46 2 (9075nn 15125 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 D are integers for the following choices of and namely, (i) s1; r  30 n  27 (ii) s9; r  30 n  7 (iii) s9; r  30 n  17 (iv) s1; r  30 n  3 (v) s11; r  30 n  3 (vi) s19; r  30 n  3 (vii) s30 n  19; r  60 n  27 (viii) s30 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 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

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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 PenP DodecD 1 1 (75n22 135 n  61)(225 n  405 n  182) (180n22 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 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 r and s namely, (i) s15 n  3; r  15 n  6 (ii) s15 m  12; r  30 n  15 m  21 (iii) s15 m  12; r  30 n  15 m  39 For each of the values of and the values of Q and H 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

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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 yx31Acta Ciencia Indica XXXIVM (2) (2008) 103-106. 22 [2] M.A.Gopalan and B.Sivakami,Observations on the integral solutions of yx71, Antarctica J. Math. 7(3),(2010) ,291- 296 22 [3] M.A.Gopalan and G.Srividhya,Relations among m-gonal numbers through equation yx21,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 yx51 , 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 yx10 1 , Impact J.Sci.Tech. 4 (2010), 103-106 22 [8] M.A.Gopalan and R.Palanikumar, Observations on yx12 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 yx10 1 , Impact J.Sci.Tech. 5(1),(2011), 57-60 22 [10] M.A.Gopalan and K.Geetha, Observations on the Hyperbola yx18 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