We analyse the Diophantine equation of Fermat xpyp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z – y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xpyp ≠ zp.
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