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Use Style: Paper Title Volume 4, Issue 11, November – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 On Three Figurate Numbers 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 Number of rank M numbers, Hexagonal numbers, Centered Hexagonal ct8,M 4MM 11 numbers, Centered Octagonal numbers, Centered Decagonal numbers and Centered Dodecagonal numbers Centered Decagonal Number of rank M satisfying the relations; ct10,M 5MM 11 t 3, N t 6,h ct 6, H , t3,N t6,h ct 8,M , t 3,N t6,h ct 10,M , t t ct 3,N 6,h 12,D are obtained. Centered Dodecagonal Number of rank D ct12,D 6DD 11 Keywords:- Equality of polygonal numbers, Centered Hexagonal numbers, Centered Octagonal numbers, Triangular Number of rank N Centered Decagonal numbers, Centered Dodecagonal NN 1 numbers, Hexagonal numbers, Triangular numbers. t3, N 2 Mathematics Subject Classification: 11D09, 11D99 Hexagonal Number 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 polygonal number simultaneously equal to an 2 2 another polygonal number has been studied. 2h h 3H 3H1 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 11 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 n1 n1 2 2 2h h 4M 4M 1 f n 5 2 6 5 2 6 , where n1 n1 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) Xn1 fn gn 2 2 6 3 3 The general solution of (5) is given by Yn1 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 n1 n1 h n1 fn gn 1 f n 3 2 2 3 2 2 , 4 2 6 where n1 n1 g n 3 2 2 3 2 2 ,n 1,0,1,... 3 3 1 Nn1 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 n1 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, Nn1 6,h n1 6,Hn1 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 t t ct , n 0, 2,4,... 3, Nn 6,h n 8,Mn n N h H t t ct n1 n1 n1 3, Nn1 6,h n1 6,Hn1 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,Ns1 6,hs1 10,Ms1 Ns1 hs1 M s1 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 5M1 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 2M1 (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 2h2 h 6D2 6D1 Ys fs , Xs gs 2 2 10 which is written as s1 s1 2 2 Y 12X 3 (11) fs 19 6 10 19 6 10 , where s1 s1 gs 19 6 10 19 6 10 ,s 1,0,1,... where Y 4h 1 , X 2D1 (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 Xs1 fs gs 2 2 10 whose general solution is given by 3 5 ~ 1 ~ 1 Y f g Yn fn , Xn gn s1 s s 2 2 10 4 3 where In view of (7) and (9), we get n1 n1 f n 7 4 3 7 4 3 , n1 n1 g 7 4 3 7 4 3 , n 1,0,1,... 1 3 5 n h f g 1 s1 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 Ns1 fs gs 4 2 10 2 1 3 X f g n1 2 n 4 n 1 1 3 M f g 1 s1 s s 3 2 2 2 10 Y f 3 g n1 2 n n Note that the values of t t ct , s 1,1,3,... 3, Ns1 6,hs1 10,Ms1 IJISRT19NOV622 www.ijisrt.com 801 Volume 4, Issue 11, November – 2019 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 In view of (10) and (12), we get 1 3 h n1 fn 3gn 1 4 2 3 3 1 N f g n1 4 n 2 n 2 1 1 3 D f g 1 n1 n n 2 2 4 Note that the values of t t ct , n 1,1,3,... 3, Nn1 6,h n1 12,Dn1 A few numerical examples satisfying the relations are given in the Table: 4 below: n t t ct N n1 h n1 Dn1 3, Nn1 6,h n1 12,Dn1 -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].
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