International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 10 Issue: 04 | Apr 2023
p-ISSN: 2395-0072
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Generation of Pythagorean triangle with Area/ Perimeter as a Wagstaff prime numbers G. Janaki1, S. Shanmuga Priya*2 1
Associate Professor, Department of Mathematics, Cauvery College for Women (Autonomous), Annamalai Nagar, Trichy, Tamil Nadu, India.
2Research Scholar, Department of Mathematics, Cauvery College for Women (Autonomous), Annamalai Nagar,
Trichy, Tamil Nadu, India. ---------------------------------------------------------------------***--------------------------------------------------------------------Definition 2: Most suitable solution of the Pythagoren Abstract - In this paper, we express the ratio
Area/Perimeter as a Wagstaff Prime number for Pythagorean triangle. Some of the interesting patterns and sides of a triangle are shown.
equation is r = m12 – m22, s= 2m1m2 and t = m12 + m22, where m1,m2 ∈ N such that m1>m2. If m1,m2 are of opposite parity and gcd(m1,m2)=1, then the solution is said to be primitive.
Key Words: Pythagorean Triangle, Wagstaff prime number.
3. WAGSTAFF PRIME NUMBER In Number Theory, Wagstaff Prime number is a prime number of the form (2p+1)/3, where p is an odd prime.
1.INTRODUCTION
The prime pages attribute the naming of the Wagstaff primes, which are named after the mathematician Samuel S. Wagstaff Jr., to Fran¸cois Morain, who did so during a lecture at the Eurocrypt 1990 conference. The New Mersenne conjecture mentions Wagstaff primes, and they are used in cryptography.
Mathematical topics include arithmetic, number theory, formulas and associated structures, shapes and the spaces in which they are contained (geometry), quantities and their changes, formulas and related structures. The Greek word ”Mathema,” which means knowledge, study, and learning, is where the word ”mathematics” originates. The area of pure mathematics known as number theory is extensively used to study integer and integer value functions. In the field of number theory, the concept of the Pythagorean triangle is a fascinating one. A wide variety of interesting problems can be found in [7, 12, 15]. In [4, 5], the authors related the Pythogorean triangle with polygonal numbers. Special types of numbers like Dhuruva numbrs, nasty numbers and Jarasandha numbers are available in [1, 2, 11, 13]. Pythogorean triangle with nasty number as a leg is discussed in [6]. Special pairs of Pythogorean triangle and rectangles with Jarasandha numbers are given in [8–10]. Common sided Pythagorean triples are discussed in [14] .In [3], connection between pythagorean triangle and dodecic number is explained. Also [16] deals with the relationship between the Pythagoren triangle and woodall primes in which the number of Pythagoren triangle generated by expressing Area/ Perimeter as woodall prime numbers.
Ryan Propper revealed the discovery of the Wagstaff probable prime 2021 which has slightly more than 4.5 million decimal digits.
4. METHOD OF ANALYSIS The symbols A1 and P1 denotes the Area and Perimeter of a Pythagoren Triangle, respectively. Assume A1/P1 = Wagstaff Prime number. This relationship results in the following equation m2(m1-m2)/2 = Wagstaff Prime number.
In this paper, we express the ratio Area/ Perimeter of a Pythogorean triangle as a Wagstaff prime number and we observe some entralling results.
2. BASIC DEFINITIONS Definition 1: Let (r, s, t) be a Pythagorean triple if it satisfies the Pythagorean equation r2 + s2 = t2 where r, s, t ∈ N. A Pythagorean triangle contains the sides as a Pythagorean triple and it is denoted by T (r, s, t) : r2 + s2 = t2 where r, s are legs and t is hypotenuse.
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