International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 09 Issue: 05 | May 2022
p-ISSN: 2395-0072
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Pythagorean Triangle with 2*A/P as Gopa Numbers Of The Second Kind S.Vidhyalakshmi1, M.A.Gopalan2 1Assistant
Professor, Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University,Trichy-620 002,Tamil Nadu, India. 2Professor, Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy-620 002, Tamil Nadu, India. ---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - This study deals with the problem of obtaining
2. Gopa numbers of the Second kind
Pythagorean triangles where, in each Pythagorean triangle,
Let N be a non-zero positive integer such that ,where and are distinct primes. If the relation Sum of the divisors of N = Product of the sum of the divisors of = square multiple of smallest nasty number holds ,then, the integer N is referred as Gopa number of the second kind
2 Area the expression Perimeter
is represented by Gopa numbers of the second kind. Also, we present the number of primitive and non-primitive Pythagorean triangles and some of the relations among them. Key Words: Pythagorean triangles, primitive Pythagorean triangle, Non-primitive Pythagorean triangle, Gopa numbers of the second kind.
Examples: 14,34,62,142,781,1067,1819
1. INTRODUCTION 3. METHOD OF ANALYSIS
It is well known that there is a one-to-one correspondence between the polygonal numbers and the sides of polygon. In addition to polygon numbers, there are other patterns of numbers namely Nasty numbers, Harshad Numbers, Dhuruva Numbers, Sphenic Numbers, Jarasandha Numbers, Armstrong Numbers and so on. In particular, refer [1-18] for Pythagorean triangles in connection with each of the above special number patterns. The above results motivated us for searching Pythagorean triangles in connection with a new number pattern. Thus, this paper exhibits Pythagorean triangles such that each Pythagorean triangle with two times the ratio Area/Perimeter is represented by a number known as Gopa numbers of the second kind. A few illustrations with the number of primitive and Non-primitive Pythagorean triangles and some of the properties involving the sides of the Pythagorean triangle are also given.
Let T x, y, z be a Pythagorean triangle, where x 2 pq, y p 2 q 2 , z p 2 q 2 , Denote the area and perimeter of respectively. The problem under consideration is
Gopa numbers of the second kind
(2)
which is equivalent to solving the binary quadratic equation given by
q p q
(3) Given , it is possible to obtain the values of and satisfying (3). Knowing and using (1), one obtains different Pythagorean triangles, each satisfying the
1. Nasty number
2 A relation P , Gopa numbers of the second kind. A
Let N be a non-zero positive integer such that N= a*b=c*d Where a, b ,c ,d are non-zero distinct integers. If the relation a+b=c-d or a-b=c+d holds, then, the integer N is referred as nasty number.
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T x, y, z by A and P
2 A P ,
2. DEFINITIONS
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p q 0 (1)
Impact Factor value: 7.529
few illustrations are presented in the Table 1 below:
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