A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation

Abstract

Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x 4 + y 4 = z 4 and x 3 + y 3 = z 3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + s p = t p where p is any prime >3 and prove the theorem by the method of contradiction. To support the proof in the above equation we have used an Auxiliary equation x 3 + y 3 = z 3 . The two equations are linked by means of transformation equations. Solving the transformation equations we prove the theorem.