V. Pandichelvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN: 2248-9622, Vol. 7, Issue 8, (Part -2) August 2017, pp.12-15

RESEARCH ARTICLE OPEN ACCESS

Relations among Some Polygonal and a Nasty of the Form Six Times a Perfect Square

*V. Pandichelvi1, P. Sivakamasundari2 1Assistant Professor, Department of Mathematics, Urumu Dhanalahshmi College, Trichy. 2Guest Lecturer, Department of Mathematics, BDUCC,Lalgudi, Trichy. Corresponding Author: V. Pandichelvi

ABSTRACT: In this paper, we discover some relations among polygonal numbers and a nasty number by applying the 2 2 2 solution of the standard equation Z  6X  Y Keywords: Ternary quadratic equation, integral solutions, polygonal numbers. ------Date of Submission: 24-07-2017 Date of acceptance: 05-08-2017 ------

NOTATIONS USED: of rank n with side m is given by  n 1m  2 Tm,n  n 1  2  I. INTRODUCTION RELATION-I Diophantine equations are numerously rich Relation among , decagonal because of its variety. Diophantine problems have number and nasty number fewer equations than unknown variables and involve Let t5,n,t10, m be the pentagonal and decagonal finding solutions in integers. There is no universal method available to know whether a Diophantine number of rank n and m respectively. equation has a solution or finding all solutions if it Assume that “two hundred and sixteen times exists. In this context one may refer [1-3]. In [4-12] pentagonal number – sixteen times decagonal 2 different types of ternary quadratic equations with number  6X , a nasty number” three unknowns are studied for its non-trivial integral 2 solutions by applying various methods. In this paper,  216t5,n 16t10, m  6X we present some relations among the polygonal 2 2 2 numbers and a nasty number by employing the  18n3 8m3  6X general solutions to the ternary quadratic equation which implies that 2 2 2 Z  6X  Y where II. METHOD OF ANALYSIS: Z  18n  3 (4) Consider the Diophantine equation Y  8m  3 (5)

From (1) and (4), we have The general solution to is given 6r 2  s2  3 by n  2 2 18 Z  6r  s (1) From (2) and (5), we get 2 2 2 2 Y  6r  s (2) 6r  s  3 m  X  2rs (3) 8 Here, we note that and m are integers when

r  6k  5, s 12k  9

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V. Pandichelvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN: 2248-9622, Vol. 7, Issue 8, (Part -2) August 2017, pp.12-15

Therefore, the values of the ranks of pentagonal and Assume that “forty times – are represented by sixteen times decagonal number , a nasty 2 n  20k  32K 13 number” 2 2 m  9k 18k  9  40t7,n 16t10, m  6X We observe that 2 2 2 2 2  10n  3  8m  3  6X

216t5, n 16t10, m  6144k  288k  9 2 2 2  Z  6X  Y Hence, 216t5,n 16t10,m is a nasty number. where RELATION-II Z  10n  3 (8) Relation among , octagonal Y  8m  3 (9) number and nasty number In view of (1) and (8), we establish that Let t3, n ,t8, m be the triangular and octagonal 6r 2  s2  3 number of rank n and m respectively. n  Assume that “eight times triangular number – three 10 2 Comparing (2) and (9), we find that times  6X , a nasty number” 2 2 2 6r  s  3  8t  3t  6X m  3,n 8, m 8  2n 12  3m 12  6X 2 Since our intension is to find the integer values for the ranks, we note that and when which can be written as r 10k 1, s  20k 1 2 2 2 Z  6X  Y Therefore, the values of the ranks of heptagonal and where decagonal numbers are expressed by Z  2n 1 2 (6) n  100k 16K 1 Y  3m 1 (7) 2 m  25k 10k 1 Equating (1) and (6), we have We observe that 2 2 6r  s 1 2 2 n  40t 16t  6 400k  60k  2 2 7, n 10, m   Comparing (2) and (7), we get Hence, 40t7, n 16t10,m is a nasty number. 2 2 6r  s 1 RELATION-IV m  3 Relation among Heptagonal number, Here, we note that the values of n and m are Hexadecagonal number and nasty number integers for the following choices of r and s Let t7,n , t16,m be the heptagonal and r  6k , s  6k 1 hexadecagonal number of rank and Therefore, the values of the ranks of triangular and respectively. octagonal number are expressed by Assume that “Forty times heptagonal number – 2 n  126k  6K Seven times hexadecagonal number , a nasty 2 number” m  60k  4k 2

We observe that  40t7,n  7t16,m  6X 2 2 2 2 2 8t3, n  3t8, m  672k 12k  10n  3  7m  3  6X which can be written as Hence, 8t3,n 3t8,m is a nasty number. 2 2 2 Z  6X  Y RELATION-III where Relation among heptagonal number, decagonal number and nasty number Z  10n  3 (10) Let be the heptagonal and decagonal Y  7m  3 (11) t7,n ,t10, m   number of rank and respectively. From (1) and (10), we find the ranks of the heptagonal number by

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V. Pandichelvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN: 2248-9622, Vol. 7, Issue 8, (Part -2) August 2017, pp.12-15

2 2 2 6r  s  3 m  450k  580k 122 n  10 We observe that From (2) and (11), we evaluate the ranks of the 2 2 hexadecagonal number by 20t12,n  9t20, m  68100k  4140k  480 2 2 6r  s  3 Hence, 20t12,n 9t20, m is a nasty number. m  7 RELATION-VI Here, we note that n and m are integers when Relation among Tridecagonal number, Icosikaidigonal number and nasty number r  35k  4 , s  70k 1 Let be the tridecagonal and Therefore, the values of the ranks of heptagonal and t13,n, t22, m hexadecagonal numbers are expressed by icosikaidigonal number of rank and 2 respectively. n  1225k 182K 10 Assume that “Eighty eight times tridecagonal 2 m  350k  220k 14 number – Forty times icosikaidigonal number , a nasty number” We observe that 2 2 2  88t13,n  40t22, m  6X 40t7,n  7t16, m  64900k  630k  8  22n 92  40m 92  6X 2 Hence, 40t  7t is a nasty number. 7,n 16,m which can be written as RELATION-V 2 2 2 Relation among Dodecagonal number, Icosagonal Z  6X  Y number and nasty number where Let be the dodecagonal and Z  22n  9 t12,n ,t20,m (14) icosagonal number of rank n and m respectively. Y  40m  9 (15) Assume that “Twenty times dodecagonal number – From (1) and (14), the ranks of the tridecagonl 2 number are given by Nine times icosagonal number  6X , a nasty 6r 2  s2  9 number” n  2 22  20t12,n  9t20, m  6X From (2) and (15), the ranks of the icosikaidigonal 2 2 2 number are noted by  10n  4  9m  4  6X which can be written as 6r 2  s2  9 2 2 2 m  Z  6X  Y 40 where Since our aim is to find the integer values for the ranks, we observe that and are integers when Z  10n  4 (12) r  440k  4 , s  880k 15 Y  9m  4 (13) Thus, the values of the ranks of tridecagonal and From (1) and (12), we find the ranks of the icosikaidigonal numbers are expressed by dodecagonal number by 2 n  88000k  2160K 15 6r 2  s2  4 n  2 10 m  9680k 132k  3 From (2) and (13), we discover the ranks of Therefore, icosagonal number by 2 2 88t13, n  40t22, m  68100k  4140k  480 6r 2  s2  4 m  Hence, 88t  40t is a nasty number. 9 13, n 22,m Here, we note that and are integers when III. CONCLUSION: r  45k 15 , s  90k 16 To conclude, one can investigate a variety of interesting relations between any other familiar Therefore, the values of the ranks of dodecagonal numbers by applying the solutions of different kinds and icosagonal numbers are pointed out by of Diophantine equations. 2 n  2025k 1098K 161

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V. Pandichelvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN: 2248-9622, Vol. 7, Issue 8, (Part -2) August 2017, pp.12-15

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V. Pandichelvi. "Relations among Some Polygonal Numbers and a Nasty Number of the Form Six Times a Perfect Square." International Journal of Engineering Research and Applications (IJERA) 7.8 (2017): 12-15.

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