D02073237

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 1

M.A.Gopalan, 2Manju Somanath, 3K.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 3 2 20 6n  n  5n deal with Icosagonal Pyramidal numbers given by pn  and various interesting relations

2

among these numbers are exhibited by means of theorems involving the relations.

Notation pnm = Pyramidal number of rank n with sides m

tm , n = Polygonal number of rank n with sides m

jaln = Jacobsthal Lucas number ctm , n = Centered Polygonal number of rank n with sides m

cpnm = Centered Pyramidal number of rank n with sides m g n = 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. 1) Proof

pn20

INTERESTING RELATIONS

 3 pn6  pn5  2t3,n



 

 

2 pn20  4n3  3n2  n  2 n3  n2  4 n2  n



 6 pn20  2 pn5  4t3, n pn20  3 pn6  pn5  2t3,n www.irjes.com

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Observations on Icosagonal Pyramidal Number

pn20  n  cpn5  pn5

2) Proof



 



2 pn20  5n3  3n  n3  n2  2n  2cpn5  2 pn5  2n

pn20  n  cpn5  pn5 pn201  pn201  2 gn  1

3) Proof

pn201  6n3  19n2  15n  2 pn201  6n3  19n2  7n  5 2 pn201  2 pn201  8n  2 pn201  pn201  2  2n  1  1

pn201  pn201  2 gn  1 4)

The following triples are in arithmetic progression 8 14 20 i) pn , pn , pn





Proof

pn20

ii)

 pn4, p12n , pn20 



pn8

6n3  n 2  5n 2n3  n 2  n   2 2  4 n3  2 n 2  6 n   2    2  

 2 p14 n

Proof

2n3  3n 2  n 6n3  n 2  5n pn4  pn20   6 2 3 2  10n  3n  14n   2    6  

iii)

 p12n , p16n , pn20 

 2 p12 n

Proof 3 2 3 2 20 10n  3n  7 n 6n  n  5n p12  p   n n 6 2  14n3  3n 2  11n   2    6  

 2 p16 n www.irjes.com

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Observations on Icosagonal Pyramidal Number





pn20  5 pn5  t3,n  0

5) Proof

     4  2 pn5   5  2t3, n 

2 pn20  2 pn5  4 n3  n2  5 n2  n

N 20 p 4 2  n  5  6 pN  t3, N n n 1

6) Proof



N 20 p 2  n   6n 2  n  5 n n 1



 6 n2   n  5 N 20 p 4 2  n  5  6 pN  t3, N n n 1

2 p 20n  2n 1 Tha2n  1  2n mern

7)

2

Proof

     

2 p20n  6 23n  22n  5 2n

2 2 p 20n 2  2 3 22 n  1  2 n  1  2 n

   

2



 2 Tha2n 1  mern

2 p 20n  2n 1 Tha2n  1  2n mern

8) i)

12



2 The following equation represents Nasty number 6 pn20  cp15 n  2 pn  n



Proof

pn20

6 n 3  n 2  5n  2



5n3  3n n3  n 2  n 2 2

6  cp15 n  2 pn  6 pn20  cp15 n  2 pn  n 

n2 n 2

n2 2

Multiply by 12, then the equation represents Nasty number 2 pn20 n5 ii)

n

Proof

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Observations on Icosagonal Pyramidal Number

2 pn20  6n3  n2  5n 2 pn20  n  5  6n 2 n pn20  n  sn  2 g n  4 n

9) Proof

pn20  6n 2  n  5 n





 6n2  6n  1  2  2n 1  n  4 pn20  n  sn  2 g n  4 n 10)



2 pn20  9n  2 nct12, n  t7, n



Proof

 6 n 2  6 n  1 5n  6  pn20  n     2 2    5n 2  3n  9n  nct12, n      2 2   9n  nct12, n  t7, n  2



2 pn20  9n  2 nct12, n  t7, n 11)



2 pn20 2  4 p11 n  t66, n  42  mod107 

Proof

2 pn20 2  6n3  37n2  69n  42



 



 2 3n3  n2  2n  35n2  34n  107n  42  4 p11 n  t66, n  107n  42 2 pn20 2  4 p11 n  t66, n  42  mod107  12)

pn203  pn203  2t56, n  29 gn  176

Proof

2 pn203  6n3  55n2  163n  156



2 pn203  6n3  53n2  151n 138

 

2 pn203  pn203  2 54n2  6n  147







pn203  pn203  54n2  52n  29  2n  1  176 pn203  pn203  2t56,n  29 gn  176 www.irjes.com

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Observations on Icosagonal Pyramidal Number N

13)





2  pn20  t3, N 6t3, N  5  sqN n 1

Proof

N

2  pn20  6 n3   n2  5 n n 1

 6t3, N  sq N  5t3, N N





2  pn20  t3, N 6t3, N  5  sqN n 1

14)

18 9 4 pn20  7n  6cpn8  cp12 n  cpn  2cpn  4t3, n

Proof

15)

2 pn20  6cpn8  cp12 n  2t3, n  4n

(1)

9 2 pn20  cp18 n  2cpn  2t3, n  3n

(2)

Add equation (1) and (2), we get 18 9 4 pn20  7n  6cpn8  cp12 n  cpn  2cpn  4t3, n 2 p 20n 2  jal 2n  2  Tha2n  car ln  mern  8 n

2

Proof

2 p 20n 2

2 n

2 p 20n 2 16)

2 n

    4  22n   1  3  22n  1   22n  2n 1 1   2n 1  8

 6 2 2 n  2n  5

 jal2n  2  Tha2n  car ln  mern  8

pn20  p14 n  n is a cubic integer

Proof

pn20  p14 n 

6n3  n 2  5n 4n3  n 2  3n  2 2

 n3  n 3 pn20  p14 n n n 17)





24 PENn  pn20  66 pn6  7n  pn is biquadratic integer

Proof





24 PENn  pn20  n4  66n3  n2  6n





 n4  66n3  (n2  n)  7n

24 PENn  pn20  66 pn6  7n  pn  n4

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