International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395 -0056
Volume: 03 Issue: 08 | Aug-2016
p-ISSN: 2395-0072
www.irjet.net
Special Pairs of Rectangles and Narcissistic Numbers of Order 3 and 4 G.Janaki1, P.Saranya2,* 1&2 Assistant
Professor,Department of Mathematics, Cauvery College for Women,Trichy-18
---------------------------------------------------------------------***--------------------------------------------------------------------Abstract - We search for infinitely many pairs of 3.METHOD OF ANALYSIS rectangles such that in each pair the sum of their areas represents the Narcissistic numbers of order 3 and 4. Let R1 ,R2 be two distinct rectangles with dimensions Also we present the number of pairs of primitive and p,q and x,y such that p = u + v, q = u - v (u > v > 0) and non-primitive rectangles. x = w + v, y = w - v (w > v >0) respectively provided Key Words: Rectangle, Narcissistic numbers of order 3 u,w,v are generators. and 4, primitive,non-primitive. Let A1 , A2 represent the areas of R1 ,R2 such that A1 + A2 equals the narcissistic numbers of order 3 and 4. 1.INTRODUCTION i.e., A1 + A2 = narcissistic numbers of order 3 and 4. The above relation which leads to the equation The ability to count dates back to prehistoric u 2 w2 2v 2 narcissist ic numbers of order 3 and 4 (1) times is evident from archaeological artifacts, such as a case(i) 10,000-year-old bone from the Congo region of Africa u 2 w 2 2v 2 153(narcissist ic number of order 3 ) (2) with tally marks scratched upon it—signs of an After performing numerical computations, it is noted unknown ancestor counting something. Very near the that there are 2 distinct values for u, v,w satisfying (2). dawn of civilization, people had grasped the idea of For simplicity and clear understanding, we have “multiplicity” and thereby had taken the first step presented the values of u, v, w, A1and A2 in Table 1. towards the study of numbers. Number theory, branch Table 1 of mathematics concerned with properties of the S.No u v w A A A +A positive integers (1, 2, 3, …). Sometimes called “higher 153 1 12 6 9 108 45 arithmetic,” it is among the oldest and most natural of mathematical pursuits. 153 2 17 14 16 93 60 For more ideas and interesting facts on numbers and special numbers like dhuruva number, nasty number It is clear that out of 2 pairs of rectangles, one pair is and jarasandha number one can refer [1-5].For Ideas non-primitive and in the other, one rectangle is on rectangles and special numbers [6-9] has been primitive and the other is non-primitive. studied. Recently in [10-12], Special Pythagorean case(ii) triangles and rectangles in connections with (3) u 2 w 2 2v 2 370(narcissist ic number of order 3 ) narcissistic numbers of order 3 and 5 are obtained. Proceeding as in case(i) we see that there are 6 distinct In this communication we search for infinitely many values for u,v,w satisfying (3) pairs of rectangles such that in each pair the sum of Table 2 their areas represents the Narcissistic numbers of S.No u v w A A A +A order 3 and 4. Also we present the number of pairs of 1 18 3 8 315 55 370 primitive and non-primitive rectangles. 2 18 7 12 275 95 370 1
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2.Basic Definitions
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Definition 1-Narcissistic Numbers: An n-digit number which is the sum of nth power of its digits is called an n-narcissistic number. It is also known as Armstrong number. Definition 2: A rectangle is said to be primitive if u, v are of opposite parity and gcd(u,v)=1, where x = u + v; y = u - v and u > v > 0.
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It is clear that out of 6 pairs of rectangles, 2 pairs are primitive and in the remaining pairs, one rectangle is primitive and the other is non-primitive. case(iii) u 2 w 2 2v 2 371 (narcissist ic number of order 3 )
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