Transcendental representation of Diophantine equation and some of its inherent properties

Abstract

We know that Diophantine equations are polynomial equations with integer coefficients and they are having integer solutions. In this paper we are revisits one of the Diophantine Equation xn + yn=znin different perspective, to study some of its inherent properties. In this paper we are proven transcendental representation of above Diophantine equations is zyn2=1+2x2-1. By substituting n = 2, the quadratic Diophantine equation is satisfies Pythagorean theorem, which is having transcendental representation zy=1+2x2-1. Also we are finding all primitive and non primitive Pythagorean triples by choosing of x value from following four disjoint Sets (whose union is becomes to Set of all positive integers). A = x,y,z:zy=1+2x2-1ifxisoddprimenumberoritspowersB = x,y,z:zy=1+2x2p-122-1ifxisoddcompositeanditspowers,forsomep=1,2,3..C = x,y,z:zy=1+2x22-1ifxisgeometricpowerof2 D=x,y,z:zy=1+2x2p22-1ifxisevencompositebutnotgeometricpowerof2,forsomep=1,2,3⋯. And with using of programming coding of ‘c’ language for above transcendental representation of Diophantine equation,we are proven Fermat’s Last Theorem for n > 2.