International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in RESEARCH ARTICLE ISSN: 2321-7758 OBSERVATIONS ON ICOSAHEDRAL NUMBER M.A.GOPALAN1, K.GEETHA2, MANJU SOMANATH3 1Department of Mathematics,Shrimati Indira Gandhi college, Trichirapalli- 02 2Department of Mathematics, Cauvery College for Women, Trichirapalli-18, 3 Department of Mathematics, National College, Trichirapalli-01, Article Received: 30/10/2013 Article Revised on: 25/11/2013 Article Accepted on:28/11/2013 ABSTRACT We obtain different relations among Icosahedral number and other two, three and four dimensional figurate numbers. INTRODUCTION The numbers that can be represented by a regular arrangement of points are called the polygonal numbers (also known as two dimensional figurate numbers). The polygonal number series can be summed to form solid three dimensional figurate numbers called Pyramidal numbers that be illustrated by pyramids[1].Numbers have varieties of patterns[2-18] and varieties of range and richness. In this n5 n2 5 n 2 communication we deal with Icosahedral numbers given by I and various interesting n 2 relations among these numbers are exhibited by means of theorems involving the relations. Keywords: Polygonal numbers, Pyramidal numbers and Special numbers 2010 MSC Classification Code: 11D99 Notation Fm,, n p = m-dimensional figurate number of rank n t = Polygonal number of rank n with sides m mn, where generated polygon is of p sides m ja = Jacobsthal number pn = Pyramidal number of rank n with sides m n jaln = Jacobsthal lucas number 395 K.GEETHA et al International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in ctmn, = Centered Polygonal number of rank n with carln = Carol number sides m mern = Mersenne number, where n is prime g = Gnomonic number of rank n with sides m n Than = Thabit ibn kurrah number pn = Pronic number INTERESTING RELATIONS 7 1) 2In 3 n 6 p n t18, n Proof: 3 2 2 2In 5 n 3 n 2 n 8 n 7 n 3 n 7 63pnn t18, n 1 10 2) I3 cp t n2 n12, n Proof: 22 2In 5 n 2 n 5 n 4 n 10 3cpnn t12, 3) 2In n 2 ct3, n ct 4, n t 22, n 10 n 0 Proof: 2 2 2 2In n 3 n 3 n 2 n 2 n 2 n 1 10 n n n2 ct3,n ct 4, n t 22, n 10 n 1 nn 4) 2I 5 ky 15(2 ) 7 2nn Proof: 2I n 2nn 2 5 2 5 2 2 2n 21n n n 5 2 2 1 15(2 ) 7 1 nn 2I 5 ky 15(2 ) 7 2nn 5) The following represents a nasty number 1 n i) 2I jal ja 3 car l mer 2n2 n 2 n n n Proof: 2I n 2nn 2 5 2 5 2 2 2n 2n 2 n 2 n n n 2 12 132 22125 396 K.GEETHA et al International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in 1 n 2I jal ja 3 car l mer 6 2n2 n 2 n n n 14 5 6 ii) 2I21n 2 p n 6 p n 33 cp n 17 n 2 Proof: 32 2I21n 40 n 40 n 14 n 2 4n3 n 2 3 n 3 n 3 n 2 33 n 3 36 n 2 17 n 2 14 5 6 2 2pn 6 p n 33 cp n 17 n 2 36 n 14 5 6 2 2I21n 2 p n 6 p n 33 cp n 17 n 2 36 n 5 iii) In73 n cp n t19, n Proof: 3 2 2 2In 5 n n 17 n 15 n 12 n 14 n 52 3cpnn t19, 6 n 7 n 52 In7 n 3 cp n t19, n 6 n 5 6) 2In2 6 cp n 2 ct 3, n ct 4, n 8 g n 29 Proof: 32 2In 2 5 n 25 n 42 n 24 3 2 2 5n n 23 n 23 n 2 2 n 2 n 1 16 n 29 5 6cpn 2 ct3, n ct 4, n 8 g n 29 7) 2In1 I n 1 3 ct 20, n 25 g n 15 0 Proof: 2 2Inn11 I 30 n 20 n 13 2 3 10n 10 n 1 25 2 n 1 15 3ct20,nn 25 g 15 5 8) Ip21nn40 1 mod7 Proof: 32 2I21n 40 n 40 n 14 n 2 5 80pnn 14 2 5 Ip21nn40 1 mod7 5 9) In n3 cp n t7, n Proof: 32 2In 5 n n 5 n 3 n 2 n 397 K.GEETHA et al International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in 5 6cpnn 2 t7, 2 5 In n3 cp n t7, n 7 10) In9 n 3 p n t18, n 4 t 3, n Proof: 7 In3 p n 4 t3, n 6 n (1) 7 2In 6 p n t18, n 3 n (2) Subtract (2) and (1), we get 7 In9 n 3 p n t18, n 4 t 3, n 58 11) 2In p n1 t 3, n 1 3 cp n t 12, n 4 mod13 Proof: 3 2 3 3 2 2In n 452 n n n 3241064 n n n n 5 3 2 2pnn1 2 t 3, 1 4 n n 10 n 8 n 13 n 4 58 2pn1 2 t 3, n 1 6 cp n 2 t 12, n 13 n 4 15 12) In cp n55 t3, n n Proof: 32 2In 5 n 3 n 5 n n 10 n 15 2cpnn 10 t3, 10 n 15 In cp n55 t3, n n 1 n 13) 2In ky n 4 car ln Mer n 8 Proof: 2I n 22n n n n n 2 2 22142 221 218 2 kynn4 car ln Mer 8 6 14) 2In 5 cpn s n p n 1 mod5 Proof: 3 2 2 2In 5 n 6 n 6 n 1 n n 5 n 1 6 5cpn s n p n 5 n 1 398 K.GEETHA et al International journal of Engineering Research-Online A Peer Reviewed International Journal Vol.1., Issue.3., 2013 Articles available online http://www.ijoer.in 57 15) 2In 3 cp n p n t7, n 8 t 3, n 5 n Proof: 5 In3 cp n t7, n n (1) 7 In3 p n 8 t3, n 6 n (2) Add (1) and (2), we get 57 2In 3 cp n p n t7, n 8 t 3, n 5 n 11 16) In23 n F4, m ,4 p n t 11, n Proof: 1 3 2 3 2 2 I2 n 3 n n 3 n n 2 n 9 n 7 n 4 n n 2 1 11 6F 2 p 2 t 4 n 2 4,m ,4 n 11, n REFERENCES [1]. 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