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An Heptagonal Numbers © April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 An Heptagonal Numbers Dr. S. Usha Assistant Professor in Mathematics, Bon Secours Arts and Science College for Women, Mannargudi, Affiliated by Bharathidasan University, Tiruchirappalli, Tamil Nadu Abstract - Two Results of interest (i)There exists an prime factor that divides n is one. First few square free infinite pairs of heptagonal numbers (푯풎 , 푯풌) such that number are 1,2,3,5,6,7,10,11,13,14,15,17…… their ratio is equal to a non –zero square-free integer and (ii)the general form of the rank of square heptagonal Definition(1.5): ퟑ ퟐ풓+ퟏ number (푯 ) is given by m= [(ퟏퟗ + ퟑ√ퟒퟎ) + 풎 ퟐퟎ Square free integer: ퟐ풓+ퟏ (ퟏퟗ − ퟑ√ퟒퟎ) +2], where r = 0,1,2…….relating to A Square – free Integer is an integer which is divisible heptagonal number are presented. A Few Relations by no perfect Square other than 1. That is, its prime among heptagonal and triangular number are given. factorization has exactly one factors for each prime that appears in it. For example Index Terms - Infinite pairs of heptagonal number, the 10 =2.5 is square free, rank of square heptagonal numbers, square-free integer. But 18=2.3.3 is not because 18 is divisible by 9=32 The smallest positive square free numbers are I. PRELIMINARIES 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23……. Definition(1.1): A number is a count or measurement heptagon: Definition(1.6): A heptagon is a seven –sided polygon. It also has Sequences of Heptagonal numbers: seven vertices, or corners where sides meet and seven The sequences of heptagonal numbers are angles. 푛 ∈ 푧 ≥ 0 begins. Definition(1.2): 0,1,7,18,34,55,81,112,148,189,235,286,342,403,469, Heptagonal numbers: 540. Heptagonal number us heptagon number. A Heptagonal number is a figurate number that represents a heptagon number is given by the II-INTRODUCTION ퟓ풏ퟐ−ퟑ풏 formula ퟐ Heptagonal number have fascinated mathematicians The first few heptagonal numbers are as they provide non-routine and challenging problems. 1,18,34,55,81,112,148,189,235,286,342,403,469,540, For a review on heptagonal numbers and their 616,697,783,874,970,1071,1177,1288,1404,1525,165 properties one may refer [1],[2]. In this 1,1782………… communication we present two results of interest relating to heptagonal numbers. Also a few relations Definition(1.3): among heptagonal and triangular numbers are Square numbers: presented. A perfect square is number that can be expressed as Result I: There exists an infinite pairs of heptagonal the product of two equal integer for example numbers (퐻푚 , 퐻푘) such that their ratio is equal to a 16,25,36….are perfect squares. non –zero square free integer. Proof: Definition (1.4): let(퐻푚 , 퐻푘) be a pair of heptagonal numbers such that Square –free numbers: 퐻 푚 =∝ ⋯ … … … … … . (1) A number is said to be square free if no prime factor 퐻푘 divides it more than once, that is largest power of a IJIRT 150954 INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 86 © April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 Where ∝ is a non –zero square-free integer. using the Where (푌0 − √∝ 푋0) standard definition on heptagonal numbers, equation 푠 푡ℎ푒 푓푢푛푑푎푚푒푛푡푎푙 푠표푙푢푡표푛 표푓 푒푞푢푎푡표푛 (1) and (1)is written as 2 (푌0 − √∝ 푋0 ) is the fundamental solution of 푌 - ∝ 2 2 푌 - ∝ 푋 = −푁 … … … … … … … (2) 푋2 = 1, In view of equation (3), the ranks of the Where N=9(∝ −1), respective heptagonal numbers are given by X=10K-3, 1 m= [(푌 − √∝ 푋 )푛+1(푌 + √∝ 푋 ) + (푌 − √∝ Y=10m-3……………………………..(3) 20 0 0 0 0 0 푛+1 The above equation (2) is well known, and it has an 푋0) (푌0 − √∝ 푋0) + 6], 1 infinite number of integral solutions. The general form 푋 = [(푌 + √∝ 푋 )푛+1(푌 + √∝ 푋 ) + (푌 − 푛 20√∝ 0 0 0 0 0 of an integral solution of the equation (1) is √∝ 푋 )푛+1(푋 √∝− 푌 ) + 6√∝], represented by 0 0 0 1 Where n=0,1,2,3,…..for the sake of simplicity a few 푌 = [(푌 + √∝ 푋 )푛+1(푌 + √∝ 푋 ) + (푌 − √∝ 푛 2 0 0 0 0 0 heptagonal numbers with their corresponding ranks 푛+1 푋0) (푌0 − √∝ 푋0)], when ∝ 2,6,7. 푎푟푒 푝푟푒푠푒푛푡푒푑 푛 푡ℎ푒 푡푎푏푙푒 퐼 푏푒푙표푤 1 푋 = [(푌 + √∝ 푋 )푛+1(푌 + √∝ 푋 ) + (푌 − √∝ 푛 2√∝ 0 0 0 0 0 푛+1 푋0) (푋0√∝ −푌0)], Values of ∝ Rank m Rank k Heptagonal number 퐻푚 Heptagonal number 퐻푘 2 72 51 12852 6426 2 2810796 1987533 19751431167846 9875715583923 2 1101188812960 77915256855 30353936253669376684560 15176968126864688342280 6 51 21 6426 1071 6 486843 198753 592539536358 98756589393 7 2214 837 122251169 1750167 7 36274357737 13710418506 3289572573025217866317 469938939003602552331 3 RESULT 2: m= [(19 + 3√40)2푟+1+(19 − 3√40)2푟+1 + 2], 20 The general form of the rank of square heptagonal where r=0,1,2,….. numbers (퐻 ) is given 푚 some example ate presented in the Table -2 3 m= [(19 + 3√40)2푟+1+(19 − 3√40)2푟+1 + 2], 20 where r=0,1,2,….. Proof: Let 퐻푚 be a square heptagonal number, We write 2 퐻푚 = 푡 … … … … … … … … (2.1) Where ‘t’ is a non –zero integer. using the definition of heptagonal number the above equation(2.1)is The rank 풎푟satisfies the recurrence relation 풎풓+ퟐ − written as 1442푚푟+1 + 푚푟 + 432 = 0. 푦2 = 40푡2 = 9 … … … … … … … . (2.2) where A Few Interesting Relation Among Heptagonal y=10m-3…………………….(2.3) Numbers: The values of ‘t’ and ‘y’ satisfying the equation (2.2) Let 퐻푚 푎푛푑 푇푆 denote the heptagonal and triangular are given by numbers respectively for the sake of simplicity and 3 brevity we present below a few interesting relations 푌 = [(19 + 3√40)푛+1+(19 − 3√40)푛+1], 푛 2 (i)퐻푚+1 + 퐻푚−1 − 2퐻푚 = 5 (Recurrence relation) 3 푛+1 푛+1 푡 = [√40(19 + 3√40) +(19 − 3√40) ] , 푠 푇푆 푛 2 (ii) ∑ 퐻 = (5푆 + 2) 푚=1 푚 3 In view of the equation (2.3), the rank m is given by 5 2 푇푠 3 3 (iii)∑푚=1 퐻푚 = [15푠 − 11푠 + 2] m = [(19 + 3√40)푛+1+ (19 − 3√40)푛+1 + 2], 6 푇 20 (iv)∑5 퐻3 = 푆 [1550푆5 − 2775푆4 + It is noted that the values of m will be an integer when 푚=1 푚 336 3 2 n is taking even values. Thus the rank m of the square 63240푆 + 101568푆 + 8948푆 − 11340] heptagonal number 퐻푚 is given by IJIRT 150954 INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 87 © April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 III.CONCLUSION One may search for the other infinite pairs of heptagonal numbers. whose ratios are non –square free integers and the other ranks of square heptagonal numbers. REFERENCE [1] M.R.Spiegel, Mathematical Handbook of formulas and Tables, Schism’s outline series,MCGraw-Hill,1968. [2] C.J.Efthimiou, Finding exact values for infinite series,Math.mag.72(1999)45-51. [3] C.Rousseau, Problems 1147, pi Mu Epsilon J.12(2007)559. [4] H.T.Davis. the summation of series principia press 1962. [5] Andrews, George E number theory Dover, New york,1971, pp: 3-4 IJIRT 150954 INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 88 .
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