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Fermat’s Last Theorem states that it is impossible to find natural numbers 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 .
We hypothesis that all r, s and t are non-zero integers in the equation
rp + sp = tp
and establish contradiction, and assert that r is a non-zero integer.
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.
By trial and error method, we create the transformed equations to the above two equations, solving by which we prove the the theorem.