Volume 4, Issue 11, November – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 On Three Figurate with Same Values

A. Vijayasankar1, Assistant Professor, Department of Mathematics, National College, Affiliated to Bharathidasan University Trichy-620 001, Tamil Nadu, India

Sharadha Kumar2, M. A. Gopalan3, Research Scholar, Professor, Department of Mathematics, Department of Mathematics, Shrimati Indira Gandhi College, National College, Affiliated to Bharathidasan University, Affiliated Bharathidasan University, Trichy-620 001, Tamil Nadu, India Trichy-620 002, Tamil Nadu, India

Abstract:- Explicit formulas for the ranks of Triangular  Centered Octagonal of rank M

numbers, Hexagonal numbers, Centered Hexagonal ct8,M  4MM 11 numbers, Centered Octagonal numbers, Centered Decagonal numbers and Centered Dodecagonal numbers  Centered of rank M satisfying the relations; ct  5MM 11 t  t  ct , t  t  ct , t  t  ct , 10,M 3, N 6,h 6, H 3,N 6,h 8,M 3,N 6,h 10,M t  t  ct 3,N 6,h 12,D are obtained.  Centered Dodecagonal Number of rank D

ct12,D  6DD 11 Keywords:- Equality of polygonal numbers, Centered Hexagonal numbers, Centered Octagonal numbers,  of rank N Centered Decagonal numbers, Centered Dodecagonal NN 1 numbers, Hexagonal numbers, Triangular numbers. t3, N  2 Mathematics Subject Classification: 11D09, 11D99  of rank h 2 I. INTRODUCTION t6,h  2h h

The theory of numbers has occupied a remarkable II. METHOD OF ANALYSIS position in the world of mathematics and it is unique among the mathematical sciences in its appeal to natural human 1. Equality of t  t  ct curiosity. Nearly every century has witnessed new and 3, N 6,h 6, H fascinating discoveries about the properties of numbers. Let N, h, H be the ranks of Triangular, Hexagonal and They form sequences, they form patterns and so on. An Centered Hexagonal numbers respectively. enjoyable topic in number theory with little need for prerequisite knowledge is polygonal numbers which is one The relation t  t of the very best and interesting subjects. A polygonal 3, N 6, h number is a number representing dots that are arranged into a geometric figure. As the size of the figure increases, the leads to number of dots used to construct it grows in a common pattern. Polygonal numbers have been meticulously studied N  2h 1 (1) since their very beginnings in ancient Greece. Numerous

discoveries arise from these peculiar polygonal numbers and The assumption t 6,h  ct6,H gives have become a popular field of research for mathematicians. In [1-5], one simultaneously equal to an 2 2 another polygonal number has been studied. 2h h  3H 3H1

The main thrust of this paper is to obtain ranks of which is written as 2 2 special polygonal and Centered polygonal numbers with the Y  6X 3 (2) same value. where  Notations Y  4h 1 , X  2H 1 (3)  Centered Hexagonal Number of rank H

ct 6, H  3H H 11

IJISRT19NOV622 www.ijisrt.com 799 Volume 4, Issue 11, November – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 To obtain the other solutions of (2), consider the pell The relation equation 2 2 Y  6X 1 t 3, N  t 6, h

whose general solution is given by leads to

~ 1 ~ 1 N  2h 1 (4) Yn  fn , Xn  gn 2 2 6 The assumption t  ct gives 6,h 8,M n1 n1 2h 2  h  4M2  4M 1 f n  5  2 6  5  2 6 , where n1 n1 g n  5  2 6  5  2 6 , n  1,0,1,... which is written as

2 2 (5) Applying Brahmagupta Lemma between X0 ,Y0  and Y  8X 1 ~ ~ X , Y , the other integer solutions of (2) are given by  n n  where

1 3 Y  4h 1 , X  2M 1 (6) Xn1  fn  gn 2 2 6 3 3 The general solution of (5) is given by Yn1  fn  gn 2 6 1 1 Yn  fn , X n  g n 2 4 2 In view of (1) and (3), we get

1  3 3  n1 n1 h n1   fn  gn 1   f n  3  2 2  3  2 2 , 4 2 6 where   n1 n1 g n  3  2 2  3  2 2 ,n  1,0,1,...

3 3 1 Nn1  fn  gn  4 2 6 2 In view of (4) and (6), we get

1 1  1 3  h n  fn  2 H   f  g 1 8 n1  n n  2  2 2 6  1 1 N  f  Note that the values of n 4 n 2 t  t  ct , n  0,1,2,... 3, Nn1 6,h n1 6,Hn1 1  1  M  4  g  A few numerical examples satisfying the relations are n  n  8  2  given in the Table: 1 below:

Note that the values of t3, N  t 6,h  ct8,M , n  0, 2,4,... n N h H t  t  ct n n n n1 n1 n1 3, Nn1 6,h n1 6,Hn1 0 13 7 6 91 A few numerical examples satisfying the relations are 1 133 67 55 8911 given in the Table: 2 below:

2 1321 661 540 873181 n N h M t  t  ct 3 13081 6541 5341 85562821 n n n 3, Nn 6,h n 8,Mn 0 1 1 1 1 4 129493 64747 52866 8384283271 2 49 25 18 1225 Table: 1:- Numerical Examples 4 1681 841 595 1413721

2. Equality of t 3, N  t 6,h  ct8,M Table 2:- Numerical Examples Let N, h, M be the ranks of Triangular, Hexagonal and 3. Equality of t  t  ct Centered Octagonal numbers respectively. 3, N 6,h 10,M Let N, h, M be the ranks of Triangular, Hexagonal and Centered Decagonal numbers respectively.

IJISRT19NOV622 www.ijisrt.com 800 Volume 4, Issue 11, November – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 The relation A few numerical examples satisfying the relations are

t 3, N  t 6, h given in the Table: 3 below:

s t  t  ct leads to 3,Ns1 6,hs1 10,Ms1 Ns1 hs1 M s1 N  2h 1 (7) - 1 1 1 1 1

The assumption t 6,h  ct10,M gives 1 2221 1111 703 2467531 3 320340 160170 101300 5.13089E+12 2h2 h  5M2 5M1 1 1 5 Table 3:- Numerical Examples which is written as

4. Equality of t 3, N  t 6,h  ct12, D Y2 10X2 1 (8) Let N, h, D be the ranks of Triangular, Hexagonal and Centered DoDecagonal numbers respectively. where Y  4h 1, X  2M1 (9) The relation

t 3, N  t 6, h To obtain the other solutions of (8), consider the pell equation leads to Y2 10X2 1 N  2h 1 (10)

whose general solution is given by The assumption t 6,h  ct12,D gives ~ 1 ~ 1 2 2 Y  f , X  g 2h h  6D 6D1 s 2 s s s 2 10 which is written as s1 s1 Y2 12X2 3 (11) fs  19  6 10   19  6 10  , where s1 s1 gs  19  6 10   19  6 10  ,s  1,0,1,... where

Y  4h 1 , X  2D1 (12)

Applying Brahmagupta Lemma between X0 ,Y0  and ~ ~ X , Y , the other integer solutions of (8) are given by To obtain the other solutions of (11), consider the pell  s s  equation 2 2 Y  12X 1 1 3 Xs1  fs  gs 2 2 10 whose general solution is given by

3 5 ~ 1 ~ 1 Y  f  g Yn  fn , Xn  gn s1 s s 2 2 10 4 3 where In view of (7) and (9), we get n1 n1 f n  7  4 3  7  4 3 ,

n1 n1 g  7  4 3  7  4 3 , n  1,0,1,... 1  3 5  n     h   f  g 1 s1  s s  4  2 10  Applying Brahmagupta Lemma between X0 ,Y0  and ~ ~ 3 5 1 Xn , Yn , the other integer solutions of (11) are given by Ns1  fs  gs  4 2 10 2 1 3 X  f  g n1 2 n 4 n 1  1 3  M   f  g 1 s1  s s  3 2  2 2 10  Y  f  3 g n1 2 n n

Note that the values of t  t  ct , s  1,1,3,... 3, Ns1 6,hs1 10,Ms1

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In view of (10) and (12), we get

1  3  h n1   fn  3gn 1 4  2 

3 3 1 N  f  g  n1 4 n 2 n 2

1  1 3  D   f  g 1 n1  n n  2  2 4 

Note that the values of t  t  ct , n  1,1,3,... 3, Nn1 6,h n1 12,Dn1

A few numerical examples satisfying the relations are given in the Table: 4 below:

n t  t  ct N n1 h n1 Dn1 3, Nn1 6,h n1 12,Dn1 -1 1 1 1 1 1 313 157 91 49141 3 60817 30409 17557 1849384153 5 11798281 5899141 3405871 6.95997E+13 Table 4:- Numerical Examples

III. CONCLUSION

This paper exhibits the ranks of three figurate numbers having the same values. As these numbers are rich in variety, one may attempt for other figurate numbers whose values are equal.

REFERENCES

[1]. M.A. Gopalan and S. Devibala, “Equality of Triangular numbers with special m-gonal numbers”, Bulletin of Allahabad Mathematical Society, 21, (2006), 25-29 [2]. M.A. Gopalan, Manju Somanath and N. Vanitha, “Equality of centered Hexagonal number with special m-gonal numbers”, Impact J.Sci.Tech, Vol.1, No.2, (2007), 31-34. [3]. M.A. Gopalan and G. Srividhya, “Observations on Centered Decagonal numbers”, Antarctica J.Math., Vol.8, No.1, (2011), 81-94. [4]. M.A. Gopalan, K. Geetha and Manju Somanath, “Equality of centered hexagonal number with special m-gonal number”, Global Journal of Mathematics and Mathematical Sciences, Vol.3, No.1, (2013), 41-45. [5]. M. A. Gopalan, Manju Somanath and K. Geetha, “Equality of with special m-gonal numbers”, Diophantus Journal of Mathematics, Vol.2, No.2, (2013), 65-70.

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