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

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. Observations on Icosagonal Pyramidal Number
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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. Observations on Icosagonal Pyramidal Number
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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. Observations on Icosagonal Pyramidal Number
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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. Observations on Icosagonal Pyramidal Number
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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. Observations on Icosagonal Pyramidal Number
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