Direct Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation

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:
rp + sp = tp
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 a Ramanujan-Nagell equation. Solving the transformed equations, we prove the theorem.