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 to support 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 is a non-zero integer; and z and z2 are irrational. 

We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations, we prove the theorem.


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  2. Washington LC. Elliptic Curves: Number Theory and Cryptography. 2nd ed. Boca Raton, London, New York: Lawrence Chapman & Hall/CRC, Taylor & Francis Group, 2003. pp. 445–48.
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