Generation of Pythagorean triangle with Area/ Perimeter as a Wagstaff prime numbers
G. Janaki1, S. Shanmuga Priya*21 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. ***
Abstract - In this paper, we express the ratio Area/PerimeterasaWagstaff PrimenumberforPythagorean triangle. Some of the interesting patterns and sides of a triangleareshown.
Key Words: Pythagorean Triangle, Wagstaff prime number
1.INTRODUCTION
Mathematicaltopicsincludearithmetic,numbertheory, formulasandassociatedstructures,shapesandthespacesin which they are contained (geometry), quantities and their changes, formulas and related structures. The Greek word ”Mathema,”whichmeansknowledge,study,andlearning,is wheretheword”mathematics”originates.Theareaofpure mathematicsknownasnumbertheoryisextensivelyusedto study integer and integer value functions. In the field of numbertheory,theconceptofthePythagoreantriangleisa fascinatingone.Awidevarietyofinterestingproblemscanbe found in [7, 12, 15]. In [4, 5], the authors related the Pythogoreantrianglewithpolygonalnumbers.Specialtypes of numbers like Dhuruva numbrs, nasty numbers and Jarasandha numbers are available in [1, 2, 11, 13]. Pythogoreantrianglewithnastynumberasalegisdiscussed in[6].SpecialpairsofPythogoreantriangleandrectangles withJarasandhanumbersaregivenin[8–10].Commonsided Pythagoreantriplesarediscussedin[14].In[3],connection between pythagorean triangle and dodecic number is explained. Also[16]dealswiththe relationshipbetweenthe Pythagorentriangleandwoodallprimesinwhichthenumber of Pythagoren triangle generated by expressing Area/ Perimeteraswoodallprimenumbers.
Inthispaper,weexpresstheratioArea/Perimeterofa Pythogorean triangle as a Wagstaff prime number and we observesomeentrallingresults
2. BASIC DEFINITIONS
Definition 1: Let (r, s, t) be a Pythagorean triple if it satisfiesthePythagoreanequation r2 + s2 = t2 where r,s,t ∈ N. A Pythagorean triangle contains the sides as a Pythagorean triple and itis denoted by T (r,s,t): r2+ s2 = t2 where r,s are legs and t is hypotenuse.
Definition 2: Most suitable solution of the Pythagoren equationisr=m12 –m22,s=2m1m2andt=m12 +m22,where m1,m2∈ Nsuchthatm1>m2.Ifm1,m2 areofoppositeparity andgcd(m1,m2)=1,thenthesolutionissaidtobeprimitive.
3. WAGSTAFF PRIME NUMBER
In Number Theory, Wagstaff Prime number is a prime numberoftheform(2p+1)/3,wherepisanoddprime.
The prime pages attribute the naming of the Wagstaff primes,whicharenamedafterthemathematician Samuel S. Wagstaff Jr., to Fran¸cois Morain, who did so during a lecture at the Eurocrypt 1990 conference. The New MersenneconjecturementionsWagstaffprimes,andtheyare usedincryptography.
Ryan Propper revealed the discovery of the Wagstaff probableprime2021whichhasslightlymorethan4.5million decimaldigits.
4. METHOD OF ANALYSIS
ThesymbolsA1andP1 denotestheAreaandPerimeter of aPythagorenTriangle,respectively.
Assume
A1/P1 =WagstaffPrimenumber.
Thisrelationshipresultsinthefollowing equation m2(m1-m2)/2=WagstaffPrimenumber.
)/2=3(onedigitWagstaffPrimenumber)
5. OBSERVATIONS
1. Thereare4Pythogoreantriangleforalltheabove 7cases,outofwhich2trianglesareprimitiveand the remaining 2 triangles are non- primitive.
2. ¼(r+s-t)isawagstaffprimenumber.
3. Out of 4 triangles in all the cases, t r is a even prime for one non-primitive triangleand cubic number for primitive triangle.
4. FortheWagstaffprimenumber3, 2(r+s−t)=Nastynumber.
5. Inallthecases,thesidess,tareconsecutive foroneoftheprimitivetriangles.
6. s+t,2(t-r)areperfectsquares.
6. CONCLUSION
In this work, generation of Pythagoren Triangles with Area/Perimeter as a Wagstaff primenumberand entrallingobservationsareshown.Further,Onemayfind thePythagorenTrianglesforanyothernumberpattern.
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