An Elementary Proof for Fermat’s Last Theorem using an Euler’s Equation

Abstract

Fermat’s  Last  Theorem  states  that  it  is  impossible  to  find  positive integers  A, B  and  C

satisfying the equation

Aⁿ  + Bⁿ  = Cⁿ

where n is any integer > 2.

Taking  the  proofs  of  Fermat  for  the  index  n =  4,  and  Euler  for  n =  3,  it  is  sufficient  to

prove the theorem for n = p, any prime > 3 [1].

We hypothesize that all r,  s and t are non-zero integers in the equation

r^p  + s^p  = t^p

and establish contradiction.

Just for supporting the proof in the above equation, we have another equation

x³  + y³  = z³

Without loss of generality, we assert that both x and y as non-zero integers; z³  a non-zero integer; z  and z²  irrational.

We create transformed equations to the above two equations through parameters, into which we have incorporated an Euler’s equation. Solving the transformed equations we prove the theorem.