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M-Gonal Number± N = Nasty Number, N = International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org M-Gonal number n = Nasty number, n = 1,2 Dr.K.Geetha Assistant Professor, Department of Mathematics, Dr.Umayal Ramanathan College for Women, Karaikudi. [email protected] Abstract: This paper concerns with study of obtaining the ranks of M-gonal numbers along with their recurrence relation such that M-gonal number n = Nasty number where n = 1,2. Key Words: Triangular number, Pentagonal number, Hexagonal number, Octagonal number and Dodecagonal number 1. INTRODUCTION may refer [3-7]. In this communication an attempt is Numbers exhibit fascinating properties, they form made to obtain the ranks of M-gonal numbers like patterns and so on [1]. In [3] the ranks of Triangular, Triangular, Pentagonal, Hexagonal, Octagonal and Pentagonal, Heptagonal, Nanogonal and Tridecagonal Dodecagonal numbers such that each of these M-gonal numbers such that each of these M-gonal number - number n = Nasty number, n = 1,2. Also the 2 = a perfect square and the ranks of Pentagonal and recurrence relations satisfied by the ranks of each of Heptagonal such that each of these M-gonal number + these M-gonal numbers are presented. 2= a perfect square are obtained. In this context one 2. METHOD OF ANALYSIS 2.1.Pattern 1: Let the rank of the nth Triangular number be A, then the identity 2 Triangular number – 2 = 6 x (1) is written as y22 48x 9 (2) where y 2A 1 (3) whose initial solution is x30 , y0 21 (4) Let xyss, be the general solution of the Pellian 22 yx48 1 (5) 1 ss11 where xs 7 48 7 48 2 48 1 ss11 ys 7 48 7 48 , s 0,1,.... 2 Applying Brahmagupta’s lemma [2] between the solutions xy00, and xyss, the sequence of values of x and y satisfying equation (2) is given by 1 ss11 xs 7 48 21 3 48 7 48 21 3 48 2 48 1 ss11 ys 7 48 21 3 48 7 48 21 3 48 , s 0,1,.... 2 Inview of (3), the rank of Triangular number is given by 3047 International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org 1 ss11 As 7 48 21 3 48 7 48 21 3 48 2 , s 0,1,2... 4 and the corresponding recurrence relation is found to be AAA194 96 0 2s 3 2 s 1 2 s 1 In a similar manner, the ranks of Pentagonal, Hexagonal, Octagonal and Dodecagonal numbers are presented in below table S.N M-Gonal General form of ranks o number 1 Pentagonal 1 ss11 number (B) Bs 5 24 7 24 5 24 7 24 2 , s 0,2,4... 12 2 Hexagonal 1 ss11 number (C) Cs 7 48 21 3 48 7 48 21 3 48 2 , s 0,2,4... 8 3 Octagonal 1 ss11 number (D) Ds 17 2 72 68 8 72 17 2 72 68 8 72 4 12 s 0,2,4... 4 Dodecagonal 1 ss11 number (E) Es 11 120 66 6 120 11 120 66 6 120 8 20 s 0,1,2... The recurrence relations satisfied by the ranks of each of these M-Gonal numbers are presented in the below table S.NO RECURRENCE RELATIONS 1 2 BBB2s 398 2 s 1 2x s 1 16 0 2 CCC2s 3194 2 s 1 2 s 1 48 0 3 DDD2s 31154 2 s 1 2 s 1 384 0 4 EEE2s 3482 2 s 1 2 s 1 192 0 2.2.Pattern 2: Assume the rank of the nth Pentagonal number to be B, then the equation Pentagonal number + 1 = 6 (6) is written as 22 yx24 23 (7) where yB61 (8) s 0,1,.... the initial solution of (7) is x0 4 , y0 19 (9) Let xyss, be the general solution of the Pellian 22 yx24 1 (10) 1 ss11 where xs 5 24 5 24 2 24 1 ss11 ys 5 24 5 24 , 2 3048 International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org On applying Brahmagupta’s lemma [2] between the solutions and the sequence of values of x and y satisfying equation (7) is given by 1 ss11 xs 5 24 19 4 24 5 24 19 4 24 2 24 1 ss11 ys 5 24 19 4 24 5 24 19 4 24 2 The rank of Pentagonal number from (8) is given by 1 ss11 Bs 5 24 19 4 24 5 24 19 4 24 2 , s 0,2,4,... 12 and the consequently recurrence relation is found to be BBB98 16 0 2s 3 2 s 1 2 s 1 On following the procedure similar to the above for Octagonal number, we get the rank and recurrence relation which are given below 1 ss11 Ds 17 2 72 8 72 17 2 72 8 72 4 , s 0,2... 12 DCD2s 31154 2 s 1 2 s 1 384 0 2.3.Pattern 3: Consider the rank of the nth Pentagonal number to be B, then the identity , Pentagonal number -2 = 6 (11) is written as 22 yx24 49 (12) 2 where x (13) whose initial solution is x0 7 , y0 35 (14) Let xyss, be the general solution of the Pellian (15) where , On using Brahmagupta’s lemmayB61 [2] between the solutions and the sequence of values of x and y satisfying equation (12) is given by s 0,1,.... 1 ss11 xs 5 24 35 7 24 5 24xy, 35 7 xy 24, 2 24 00 ss 1 yx22ss1124 1 ys 5 24 35 7 24 5 24 35 7 24 21ss11 xs 5 24 5 24 Inview of (13),2 the 24 ranks of Pentagonal number is given by 1 ss11 Bs 1 5 24ss11 35 7 24 5 24 35 7 24 2 , s 1,3,5,.... ys 125 24 5 24 2 3049 International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org and the corresponding recurrence relation is found to be On applying the same procedure to Octagonal number, the rank and recurrence relation are found which are given below 1 ss11 Ds 17 2 72 10 72 17 2 72 10 72 4 , s 1,3,5,.... 12 2.4.Pattern 4: Suppose the rank of the nth Pentagonal number to be B, then the identity, Pentagonal number +2 = 6 (16) is written as BBB2s 398 2 s 1 2 s 1 16 0 22 yx24 47 (17) where (18) whose initial solution is x0 2 , y0 7 (19) Let xyss, be the general solution of the Pellian DCD2s 31154 2 s 1 2 s 1 384 0 (20) where , x2 By means of Brahmagupta’s lemma [2] among the solutions of and the sequence of values of x and y satisfying equation (17) is given by 1 ss11 xs 5 24 7224 5 24 7224 2 24 1 ss11 ys 5 24 7224 5 24 7224 2 Therefore from (18), the rank of Pentagonal number is given by 1 ss11 Bs 5 24 7224 5 24 7224 2 , s 0,2,4,.... 12 yB61 and the corresponding recurrence relation is found to bes 0,1,.... 3. CONCLUSION REFERENCE xy , xy, In this paper the ranks of M-gonal numbers such that M- [1] L.E.Dikson00(1952): “Historyss of Theory of gonal number ± n = nasty number, n=1,2.yx22 In24 this manner 1 Numbers”, Chelsa Publishing Company, New one can scrutinize the ranks of M-gonal number York, 2. satisfying various properties1 along withss 11recurrence [2] T.S.Bhanumathy(1995): Ancient Indian relation. xs 5 24 5 24 Mathematics, New Age Publishers Ltd, New Delhi, 2 24 1995. 1 ss11[3] V.E. Hoggatt Jr, Fibonacci and Lucas numbers, ys 5 24 5 24 2 Houghton-Mifflin, Boston 1969. 3050 International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org [4] B.L. Bhatia, and Mohanty, Supriya , Nasty numbers and their characterization, Mathematical Education, II (1), 1985,34-37. [5] M.A.Gopalan., K.Geetha and Manju Somanath (Oct- Nov 2014): M-gonal 2 = A perfect square, International journal of Innovative Technology and Research, Vol.2, Issue 6, Pp.1618-1620. [6] M.A.Gopalan., K.Geetha and Manju Somanath, “Equality of centered hexagonal number with special M-gonal number”, Global journal of Mathematics and Mathematical Science, Vol.3, No.1, PP.41-45, 2013. [7] M.A.Gopalan., K.Geetha and Manju Somanath, M- gonal n = A perfect square, n= 5,17, Jamal Academic research journal: An interdisciplinary special issue, Pp.259-264, February 2016. 3051 .
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