We obtain the ranks of m-gonal numbers such that the difference between any two m-gonal numbers is unity. The recurrence relations satisfied by the ranks of each m-gonal numbers are also presented.

1. IJESM Volume 2, Issue 2 ISSN: 2320-0294
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2013
On Pairs of M- Gonal numbers with unit
difference
M.A.Gopalan
Manju Somanath
K.Geetha
Abstract
We obtain the ranks of m-gonal numbers such that the difference between any two
m-gonal numbers is unity. The recurrence relations satisfied by the ranks of each m-gonal
numbers are also presented.
Introduction
Number is the essence of mathematical calculation. Variety of numbers has variety of
range and richness. Many numbers exhibit fascinating properties, they form sequences, they
form patterns and so on [1,2,3]. In [4] explicit formulas for the ranks of Triangular numbers
which simultaneously equal to Pentagonal, Octagonal, Decagonal and Dodecagonal numbers in
turn are presented. Denoting the ranks of Triangular, Pentagonal, Hexagonal, Octagonal,
Heptagonal, Decagonal and Dodecagonal number by the symbols N, P, Q, M, H, D and T
respectively. In [5], the following relations are studied:
1. N - P = 1 2.N - M = 1 3.N - H = 1 4.N – D = 1
5. N – T =1 6.P – M =1 7.P – Q =1 8.P – H = 1
In [6], the ranks of m-gonal numbers such that the difference between any two m-gonal numbers
is unity. The recurrence relations satisfied by the ranks of each m-gonal number in turn are

Department of Mathematics, Shrimati Indira Gandhi College, Tiruchirapalli

Department of Mathematics, National College, Tiruchirapalli

Department of Mathematics, Cauvery College for Women, Tiruchirapalli

2. IJESM Volume 2, Issue 2 ISSN: 2320-0294
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June
2013
presented. In this communication, we make an attempt to obtain the ranks of other pairs of m-
gonal numbers such that the difference between any two m-gonal numbers is unity. The
recurrence relations satisfied by the ranks of m-gonal number are also presented.
Method of Analysis
1. Tridecagonal number – Triangular number = 1
Denoting the ranks of the Tridecagonal number and Triangular number to be A and N
respectively, the identify
Tridecagonal number – Triangular number = 1 (1)
is written as
2 2
11 158y x  (2)
where 2 1x N  , 22 9y A  (3)
whose intial solutions is 0 1x  , 0 13y 
Let ( , )n nx y be the general solution of the Pellian 2 2
11 1y x 
   
1 11
10 3 11 10 3 11
2 11
s s
sx
 
    
  

   
1 11
10 3 11 10 3 11
2
s s
sy
 
    
  
 , 0,1,2,...s 
Applying Brahmagupta’s lemma between the solutions 0 0( , )x y and ( , )n nx y the sequence
of values of x and y satisfying equation (2) is given by
       
1 11
10 3 11 13 11 10 3 11 13 11
2 11
s s
sx
 
      
  
       
1 11
10 3 11 13 11 10 3 11 13 11
2
s s
sy
 
      
  
, 0,1,2,...s 
In view of (3), the ranks of Tridecagonal and Triangular numbers are respectively given
by
       
2 2
2 1
1
10 3 11 13 11 10 3 11 13 11 18
44
s s
sA 
       
  

3. IJESM Volume 2, Issue 2 ISSN: 2320-0294
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2013
       
2 2
2 1
1
10 3 11 13 11 10 3 11 13 11 2 11
4 11
s s
sN 
       
  
,
1,2,...s 
and their corresponding recurrence relations are found to be
2 3 2 1 2 1398 162s s sA A A      , 1 3148, 58741A A 
2 3 2 1 2 1398 162s s sN N N      , 1 3489, 194820N N 
In similar manner, we present below the ranks of other pairs of m-gonal numbers in a
tabular form
S.NO M-Gonal
number
General form of Ranks
1 Heptadecagonal
number(E)
Triangular
number(N)
       
2 2
2 1
1
4 15 17 15 4 15 17 15 26
60
s s
sE 
       
  
       
2 2
2 1
1
4 15 17 15 4 15 17 15 2 15
4 15
s s
sN 
       
  
1,2,...s 
2 Nanodecagonal
number(F)
Triangular
number(N)
       
2 2
2 1
1
33 8 17 19 17 33 8 17 19 17 30
68
s s
sF 
       
  
       
2 2
2 1
1
33 8 17 19 17 33 8 17 19 17 2 17
4 17
s s
sN 
       
  
1,2,...s 
3 Nanogonal
number(M)
Triangular
number(N)
       
1 11
8 3 7 9 7 8 3 7 9 7 10
28
s s
sM
 
       
  
       
1 11
8 3 7 9 7 8 3 7 9 7 2 7
4 7
s s
sN
 
       
  
0,1,2,...s 
4 Nanogonal
number(M)
Hexagonal
number(H)
       
1 11
8 3 7 9 7 8 3 7 9 7 10
28
s s
sM
 
       
  
       
1 11
8 3 7 9 7 8 3 7 9 7 2 7
8 7
s s
sH
 
       
  
0,1,2,...s 
The recurrence relations satisfied by the ranks of each of the m-gonal numbers presented
in the table above are as follows

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S.No Recurrence Relations
1 2 3 2 1 2 162 26s s sE E E      , 1 322, 1337E E 
2 3 2 1 2 162 26s s sN N N      , 1 383, 5176N N 
2 2 3 2 1 2 14354 1920s s sF F F      , 1 31481, 6446353F F 
2 3 2 1 2 14354 1920s s sN N N      , 1 36104, 26578992N N 
3 2 116 140s s sM M M     , 0 17, 106M M 
2 116 140s s sN N N     , 0 117, 279N N 
4 2 116 140s s sM M M     , 0 17, 106M M 
2 116 140s s sH H H     , 0 19, 140H H 
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3. Shailesh Shirali, “Mathematical Marvels”, A Primer on Number Sequences, Universities Press,
2001.
4. Gopalan M.A., and Devibala S., “Equality of Triangular Numbers with Special m-gonal
numbers”, Bulletin of the Allahabad Mathematical Society, Vol.21, Pp.25-29, 2006.
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