International Refereed Journal of Engineering and Science (IRJES)
ISSN (Online) 2319-183X, (Print) 2319-1821
Volume 2, Iss...
Observations on Icosagonal Pyramidal Number
www.irjes.com 33 | Page
2)
20 5 5
n n np n cp p  
Proof
   20 3 3 2
2 5...
Observations on Icosagonal Pyramidal Number
www.irjes.com 34 | Page
5)  20 5
3,5 0n n np p t  
Proof
   20 5 3 2 ...
Observations on Icosagonal Pyramidal Number
www.irjes.com 35 | Page
20 3 2
2 6 5np n n n  
20
22
5 6np
n n
n
  
9)
2...
Observations on Icosagonal Pyramidal Number
www.irjes.com 36 | Page
13)  20
3, 3,
1
2 6 5
N
n N N N
n
p t t sq

  
...
Observations on Icosagonal Pyramidal Number
www.irjes.com 37 | Page
REFERENCES
[1]. Beiler A. H., Ch.18 in Recreations in ...
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International Refereed Journal of Engineering and Science (IRJES)

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International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications

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International Refereed Journal of Engineering and Science (IRJES)

  1. 1. International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 7 (July 2013), PP. 32-37 www.irjes.com www.irjes.com 32 | Page Observations On Icosagonal Pyramidal Number 1 M.A.Gopalan, 2 Manju Somanath, 3 K.Geetha, 1 Department of Mathematics, Shrimati Indira Gandhi college, Tirchirapalli- 620 002. 2 Department of Mathematics, National College, Trichirapalli-620 001. 3 Department of Mathematics, Cauvery College for Women, Trichirapalli-620 018. ABSTRACT:- We obtain different relations among Icosagonal Pyramidal number and other two, three and four dimensional figurate numbers. Keyword:- Polygonal number, Pyramidal number, Centered polygonal number, Centered pyramidal number, Special number MSC classification code: 11D99 I. INTRODUCTION Fascinated by beautiful and intriguing number patterns, famous mathematicians, share their insights and discoveries with each other and with readers. Throughout history, number and numbers [2,3,7-15] have had a tremendous influence on our culture and on our language. Polygonal numbers can be illustrated by polygonal designs on a plane. The polygonal number series can be summed to form “solid” three dimensional figurate numbers called Pyramidal numbers [1,4,5 and 6] that can be illustrated by pyramids. In this communication we deal with Icosagonal Pyramidal numbers given by 3 2 20 6 5 2 n n n n p    and various interesting relations among these numbers are exhibited by means of theorems involving the relations. Notation m np = Pyramidal number of rank n with sides m ,m nt = Polygonal number of rank n with sides m njal = Jacobsthal Lucas number ,m nct = Centered Polygonal number of rank n with sides m m ncp = Centered Pyramidal number of rank n with sides m ng = Gnomonic number of rank n with sides m np = Pronic number ncarl = Carol number nmer = Mersenne number, where n is prime ncul = Cullen number nTha = Thabit ibn kurrah number II. INTERESTING RELATIONS 1) 20 6 5 3,3 2n n n np p p t   Proof      20 3 2 3 2 2 2 4 3 2 4np n n n n n n n       20 5 3,6 2 4n n np p t   20 6 5 3,3 2n n n np p p t  
  2. 2. Observations on Icosagonal Pyramidal Number www.irjes.com 33 | Page 2) 20 5 5 n n np n cp p   Proof    20 3 3 2 2 5 3 2np n n n n n     5 5 2 2 2n ncp p n   20 5 5 n n np n cp p   3) 20 20 1 1 2 1n n np p g    Proof 20 3 2 1 6 19 15 2np n n n     20 3 2 1 6 19 7 5np n n n     20 20 1 12 2 8 2n np p n     20 20 1 1 2 2 1 1n np p n     20 20 1 1 2 1n n np p g    4) The following triples are in arithmetic progression i)  8 14 20 , ,n n np p p Proof 3 2 3 2 20 8 6 5 2 2 2 n n n n n n n n p p        3 2 4 2 6 2 2 n n n          14 2 np ii)  4 12 20 , ,n n np p p Proof 3 2 3 2 4 20 2 3 6 5 6 2 n n n n n n n n p p        3 2 10 3 14 2 6 n n n          12 2 np iii)  12 16 20 , ,n n np p p Proof 3 2 3 2 12 20 10 3 7 6 5 6 2 n n n n n n n n p p        3 2 14 3 11 2 6 n n n          16 2 np
  3. 3. Observations on Icosagonal Pyramidal Number www.irjes.com 34 | Page 5)  20 5 3,5 0n n np p t   Proof    20 5 3 2 2 2 2 4 5n np p n n n n        5 3,4 2 5 2n np t  6) 20 4 3, 1 2 5 6 N n N N n p p t n    Proof   20 2 1 2 6 5 N n n p n n n     2 6 5n n    20 4 3, 1 2 5 6 N n N N n p p t n    7)  20 1 2 2 2 2 1 2n n n n np Tha mer    Proof      20 3 2 2 2 6 2 2 5 2n n n n p         20 22 2 2 3 2 1 2 1 2 2 n n n n p       22 1n nTha mer    20 1 2 2 2 2 1 2n n n n np Tha mer    8) The following equation represents Nasty number i)  20 15 6 12 2n n np cp p n   Proof 3 2 20 6 5 2 n n n n p    3 3 2 5 3 2 2 n n n n n      2 15 6 2 2 n n n cp p n    2 20 15 6 2 2 n n n n p cp p n    Multiply by 12, then the equation represents Nasty number ii) 20 2 5np n n   Proof
  4. 4. Observations on Icosagonal Pyramidal Number www.irjes.com 35 | Page 20 3 2 2 6 5np n n n   20 22 5 6np n n n    9) 20 2 4n n n p n s g n     Proof 20 2 6 5np n n n       2 6 6 1 2 2 1 4n n n n       20 2 4n n n p n s g n     10)  20 12, 7,2 9 2n n np n nct t   Proof 2 20 6 6 1 5 6 2 2 n n n n p n            2 12, 5 3 9 2 2 n n n n nct           12, 7, 9 2 n n n nct t    20 12, 7,2 9 2n n np n nct t   11)  20 11 2 66,2 4 42 mod107n n np p t    Proof 20 3 2 22 6 37 69 42np n n n        3 2 2 2 3 2 35 34 107 42n n n n n n       11 66,4 107 42n np t n     20 11 2 66,2 4 42 mod107n n np p t    12) 20 20 3 3 56,2 29 176n n n np p t g     Proof 20 3 2 32 6 55 163 156np n n n     20 3 2 32 6 53 151 138np n n n        20 20 2 3 32 2 54 6 147n np p n n        20 20 2 3 3 54 52 29 2 1 176n np p n n n       20 20 3 3 56,2 29 176n n n np p t g    
  5. 5. Observations on Icosagonal Pyramidal Number www.irjes.com 36 | Page 13)  20 3, 3, 1 2 6 5 N n N N N n p t t sq     Proof 20 3 2 1 2 6 5 N n n p n n n        3, 3,6 5N N Nt sq t    20 3, 3, 1 2 6 5 N n N N N n p t t sq     14) 20 8 12 18 9 3,4 7 6 2 4n n n n n np n cp cp cp cp t      Proof 20 8 12 3,2 6 2 4n n n np cp cp t n    (1) 20 18 9 3,2 2 2 3n n n np cp cp t n    (2) Add equation (1) and (2), we get 20 8 12 18 9 3,4 7 6 2 4n n n n n np n cp cp cp cp t      15) 20 2 2 2 2 n 2 l 8 2 n n n nn p jal Tha car mer     Proof   20 22 2 6 2 2 5 2 n n n n p             2 2 2 1 4 2 1 3 2 1 2 2 1 2 1 8n n n n n           20 2 2 2 2 n 2 l 8 2 n n n nn p jal Tha car mer     16) 20 14 n np p n  is a cubic integer Proof 3 2 3 2 20 14 6 5 4 3 2 2 n n n n n n n n p p        3 n n  20 14 3 n np p n n   17)  20 6 24 66 7n n n nPEN p p n p    is biquadratic integer Proof  20 4 3 2 24 66 6n nPEN p n n n n     4 3 2 66 ( ) 7n n n n n      20 6 4 24 66 7n n n nPEN p p n p n    
  6. 6. Observations on Icosagonal Pyramidal Number www.irjes.com 37 | Page REFERENCES [1]. Beiler A. H., Ch.18 in Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, New York, Dover, (1996), 184-199. [2]. Bert Miller, Nasty Numbers, The Mathematics Teachers, 73(9), (1980), 649. [3]. Bhatia B.L., and Mohanty Supriya, (1985) Nasty Numbers and their characterization, Mathematical Education, I (1), 34-37. [4]. Bhanu Murthy T.S., (1990)Ancient Indian Mathematics, New Age International Publishers Limited, New Delhi. [5]. Conway J.H., and Guy R.K., (1996) The Book of Numbers, New York Springer – Verlag, 44-48. [6]. Dickson L.E., (1952), History of the Numbers, Chelsea Publishing Company, New York. [7]. Gopalan M.A., and Devibala S., (2007), “On Lanekal Number”, Antarctica J. Math, 4(1), 35-39. [8]. Gopalan M.A., Manju Somanath, and Vanitha N., (2007), “A Remarkable Lanekel Sequence”, Proc. Nat. Acad. Sci. India 77(A), II,139-142. [9]. Gopalan M.A., Manju Somanath, and Vanitha N., (2007),“On R2 Numbers”, Acta Ciencia Indica, XXXIIIM(2), 617-619. [10]. Gopalan M.A., and Gnanam A., (2008), “Star Numbers”, Math. Sci. Res.J., 12(12), 303-308. [11]. Gopalan M.A., and Gnanam A.,(2009),“Magna Numbers”, Indian Journal of Mathematical Sciences, 5(1), 33-34. [12]. Gopalan M.A., and Gnanam A., (2008) “A Notable Integer Sequence”, Math. Sci.Res. J., 1(1) 7-15. [13]. Horadam A.F., (1996), Jacobsthal Representation Numbers, Fib. Quart., 34, 40-54. [14]. Meyyappan M., (1996), Ramanujan Numbers, S.Chand and Company Ltd., First Edition. [15]. Shailesh Shirali, (1997), Primer on Number Sequences, Mathematics Student, 45(2), 63-73.

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