Transcendental representation of Diophantine Equation xn + yn=zn to Generate At most All Pythagorean and Reciprocal Pythagorean Triples

Abstract

This paper revisits one of the Diophantine Equations xn + yn=zn And its transcendental representation

 zyn2=1+2x2-1. By substituting n=2, the quadratic Diophantine equation satisfies Pythagorean Theorem. This paper introduced to a generation of all primitive and Nonprimitive Pythagorean triples for each positive integer ‘x ‘. By substituting n = -2 in the above Diophantine equation it satisfies the Reciprocal Pythagorean Theorem 1x2+1y2=1z2. Also verified each Pythagorean Triple (a, b, c) is generates Reciprocal Pythagorean Triple ac,bc,ab .

Apply this corollary to generate Set of Reciprocal Pythagorean Triples RPT=.a ,b, c:1a2+1b2=1c2. Also, verified each p=(a, b, c)  RPT, h=abc  (Also, c = abh );  h = cacb ; a2=ca.c ; b2=cb.c; c = 2R; r = a+b-c2 ;