Observations on Icosagonal Pyramidal Number

Home , Carol number, Cullen number, Pronic number, Pyramidal number

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) pn32 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

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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) pn11 p n 21 g n  Proof 20 3 2 pn1 6 n  19 n  15 n  2 20 3 2 pn1 6 n  19 n  7 n  5 20 20 2pnn11 2 p  8 n  2 20 20 pnn11 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

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20 5 5) pn50 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, n1 n Proof N 20 p 2 2n  6nn   5 n1 n 2 65nn  

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   21Tha2nn   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

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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) 2pn2 4 p n  t 66, n  42 mod107 Proof 20 3 2 2pn2  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 2pn2 4 p n  t 66, n  42 mod107 20 20 12) pn3 p n 3 2 t 56, n  29 g n  176 Proof 20 3 2 2pn3  6 n  55 n  163 n  156 20 3 2 2pn3  6 n  53 n  151 n  138 20 20 2 2 pnn33 p  2 54 n  6 n  147 20 20 2 pnn33 p 54 n  52 n  29 2 n  1  176

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N 20 13) 2 pn t3, N 6 t 3, N  5  sq N n1 Proof N 20 3 2 2pn  6  n   n  5  n n1 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) 24PENn p n  66 p n  7 n  p n is biquadratic integer Proof 20 4 3 2 24PENnn p  n  66 n  n  6 n 4 3 2 n 66 n  ( n  n )  7 n 20 6 4 24PENn p n  66 p n  7 n  p n  n

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