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 Observations On Icosagonal Pyramidal Number 1M.A.Gopalan, 2Manju Somanath, 3K.Geetha, 1Department of Mathematics, Shrimati Indira Gandhi college, Tirchirapalli- 620 002. 2Department of Mathematics, National College, Trichirapalli-620 001. 3Department 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 32 20 65n n n deal with Icosagonal Pyramidal numbers given by p and various interesting relations n 2 among these numbers are exhibited by means of theorems involving the relations. Notation m pn = Pyramidal number of rank n with sides m tmn, = Polygonal number of rank n with sides m jaln = Jacobsthal Lucas number ctmn, = Centered Polygonal number of rank n with sides m m cpn = Centered Pyramidal number of rank n with sides m gn = Gnomonic number of rank n with sides m pn = Pronic number carln = Carol number mern = Mersenne number, where n is prime culn = Cullen number Than = Thabit ibn kurrah number II. INTERESTING RELATIONS 20 6 5 1) pn32 p n p n t3, n Proof 20 3 2 3 2 2 2pn 4 n 3 n n 2 n n 4 n n 20 5 6pn 2 p n 4 t3, n www.irjes.com 32 | Page Observations on Icosagonal Pyramidal Number 20 5 5 2) pn n cp n p n Proof 20 3 3 2 2pn 5 n 3 n n n 2 n 55 2cpnn 2 p 2 n 20 5 5 pn n cp n p n 20 20 3) pn11 p n 21 g n Proof 20 3 2 pn1 6 n 19 n 15 n 2 20 3 2 pn1 6 n 19 n 7 n 5 20 20 2pnn11 2 p 8 n 2 20 20 pnn11 p 2 2 n 1 1 4) The following triples are in arithmetic progression 8 14 20 i) pn,, p n p n Proof 3 2 3 2 20 8 6n n 5 n 2 n n n pp nn 22 4n32 2 n 6 n 2 2 14 2 pn 4 12 20 ii) pn,, p n p n Proof 3 2 3 2 4 20 2n 3 n n 6 n n 5 n pp nn 62 10n32 3 n 14 n 2 6 12 2 pn 12 16 20 iii) pppnnn,, Proof 3 2 3 2 12 20 10n 3 n 7 n 6 n n 5 n pp nn 62 14n32 3 n 11 n 2 6 16 2 pn www.irjes.com 33 | Page Observations on Icosagonal Pyramidal Number 20 5 5) pn50 p n t3, n Proof 20 5 3 2 2 2pnn 2 p 4 n n 5 n n 5 4 2ptnn 5 2 3, N 20 pn 4 6) 2 5 6 ptNN 3, n1 n Proof N 20 p 2 2n 6nn 5 n1 n 2 65nn 20nn 1 7) 2p 2 Tha2nn 1 2 mer 2n Proof 20 3n 2 n n 2p 6 2 2 5 2 2n 2p20 n 2nn 2 2 3 2 1 2 1 2 2n 21Tha2nn mer 8) The following equation represents Nasty number 20 15 6 i) 12 pn cp n 2 p n n Proof 32 20 65n n n p n 2 53n3 n n 3 n 2 n 22 2 15 6 n cp 2 p n nn2 2 20 15 6 n p cp 2 p n n n n 2 Multiply by 12, then the equation represents Nasty number 2 p20 ii) n n 5 n Proof www.irjes.com 34 | Page Observations on Icosagonal Pyramidal Number 20 3 2 2pn 6 n n 5 n 20 2 p 2 n nn 56 n p20 9) n n s 24 g n nn Proof 20 p 2 n 65nn n 2 6n 6 n 1 2 2 n 1 n 4 p20 n n s 24 g n nn 20 10) 2pn 9 n 2 nct12, n t 7, n Proof 2 20 6n 6 n 1 5 n 6 pn n 22 5n2 3 n 9 n nct 12,n 22 9n nct t 12,nn 7, 2 20 2pn 9 n 2 nct12, n t 7, n 20 11 11) 2pn2 4 p n t 66, n 42 mod107 Proof 20 3 2 2pn2 6 n 37 n 69 n 42 3 2 2 2 3n n 2 n 35 n 34 n 107 n 42 11 4pnn t66, 107 n 42 20 11 2pn2 4 p n t 66, n 42 mod107 20 20 12) pn3 p n 3 2 t 56, n 29 g n 176 Proof 20 3 2 2pn3 6 n 55 n 163 n 156 20 3 2 2pn3 6 n 53 n 151 n 138 20 20 2 2 pnn33 p 2 54 n 6 n 147 20 20 2 pnn33 p 54 n 52 n 29 2 n 1 176 www.irjes.com 35 | Page Observations on Icosagonal Pyramidal Number N 20 13) 2 pn t3, N 6 t 3, N 5 sq N n1 Proof N 20 3 2 2pn 6 n n 5 n n1 65t3,NNN sq t 3, 20 8 12 18 9 14) 4pn 7 n 6 cp n cp n cp n 2 cp n 4 t3, n Proof 20 8 12 2pn 6 cp n cp n 2 t3, n 4 n (1) 20 18 9 2pn cp n 2 cp n 2 t3, n 3 n (2) Add equation (1) and (2), we get 2p20 2n 15) jal2n 2 Tha 2 n carl8 n mer n 2n Proof 2p20 n 2nn 2 6 2 2 5 2n 2n 2 n 2 n n 1 n 42 132 12 2 1218 20 14 16) pnn p n is a cubic integer Proof 3 2 3 2 20 14 6n n 5 n 4 n n 3 n pp nn 22 3 nn 20 14 3 pnn p n n 20 6 17) 24PENn p n 66 p n 7 n p n is biquadratic integer Proof 20 4 3 2 24PENnn p n 66 n n 6 n 4 3 2 n 66 n ( n n ) 7 n 20 6 4 24PENn p n 66 p n 7 n p n n www.irjes.com 36 | Page Observations on Icosagonal Pyramidal Number REFERENCES [1]. 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